opennlp-wsd/src/test/resources/semcor3.0/brown1/tagfiles/br-j18

<contextfile concordance="brown">
<context filename="br-j18" paras="yes">
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<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VBP" ot="notag">are</wf>
<wf cmd="done" pos="VB" lemma="try" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1300736.001">trying</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="study" wnsn="1" lexsn="2:31:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295106.015">study</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="finite" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247165.001">finite</wf>
<wf cmd="done" pos="JJ" lemma="dimensional" wnsn="2" lexsn="5:00:00:multidimensional:00">dimensional</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="done" pos="VB" lemma="decompose" wnsn="1" lexsn="2:30:00::">decomposing</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">into</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" ot="notag">direct_sum</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operators</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">some</wf>
<wf cmd="done" pos="NN" lemma="sense" wnsn="0" lexsn="1:09:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1289077.007">sense</wf>
<wf cmd="done" pos="JJ" lemma="elementary" wnsn="1" lexsn="5:00:00:easy:01">elementary</wf>
<punc>.</punc>
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<s snum="2">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="done" pos="VB" lemma="do" wnsn="1" lexsn="2:41:01::">do</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="ignore" pos="IN">through</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="value" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303448.007">values</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vectors</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="JJ" lemma="certain" wnsn="1" lexsn="5:00:00:definite:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231804.003">certain</wf>
<wf cmd="done" pos="JJ" lemma="special" wnsn="1" lexsn="5:00:01:specific:00">special</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="2" lexsn="1:26:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">cases</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<wf cmd="ignore" pos="WRB">when</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="factor" wnsn="1" lexsn="2:31:00::">factors</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="scalar_field" wnsn="1" lexsn="1:14:00::">scalar_field</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<wf cmd="ignore" pos="IN">into</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="product" wnsn="3" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280957.006">product</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="JJ" lemma="distinct" wnsn="1" lexsn="5:00:00:different:00">distinct</wf>
<wf cmd="done" pos="NN" lemma="monic_polynomial" wnsn="1" lexsn="1:09:00::">monic_polynomials</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::">degree</wf>
<wf cmd="done" pos="NN" lemma="1" wnsn="1" lexsn="1:23:00::">1</wf>
<punc>.</punc>
</s>
<s snum="3">
<wf cmd="ignore" pos="WP">What</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="do" wnsn="1" lexsn="2:41:01::">do</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="general" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1250413.001">general</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>?</punc>
</s>
<s snum="4">
<wf cmd="ignore" pos="IN">If</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="try" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1300736.001">try</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="study" wnsn="1" lexsn="2:31:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295106.015">study</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="use" wnsn="1" lexsn="2:34:01::">using</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="value" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303448.007">values</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VBP" ot="notag">are</wf>
<wf cmd="done" pos="VB" lemma="confront" wnsn="3" lexsn="2:32:03::">confronted</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="JJ" lemma="two" wnsn="1" lexsn="5:00:00:cardinal:00">two</wf>
<wf cmd="done" pos="NN" lemma="problem" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280894.001">problems</wf>
<punc>.</punc>
</s>
<s snum="5">
<wf cmd="done" pos="RB" lemma="first" wnsn="1" lexsn="4:02:00::">First</wf>
<punc>,</punc>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" ot="notag">may</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.004">not</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="2" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="single" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1290881.001">single</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="value" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303448.007">value</wf>
<punc>;</punc>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="really" wnsn="1" lexsn="4:02:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1283295.001">really</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="deficiency" wnsn="1" lexsn="1:26:00::">deficiency</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="scalar_field" wnsn="1" lexsn="1:14:00::">scalar_field</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="namely" wnsn="1" lexsn="4:02:00::">namely</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.001">not</wf>
<wf cmd="done" pos="RB" lemma="algebraically" wnsn="1" lexsn="4:02:00::">algebraically</wf>
<wf cmd="done" pos="JJ" lemma="closed" wnsn="2" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233818.001">closed</wf>
<punc>.</punc>
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<s snum="6">
<wf cmd="done" pos="RB" lemma="second" wnsn="1" lexsn="4:02:00::">Second</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="even" wnsn="1" lexsn="4:02:00::">even</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="VB" lemma="factor" wnsn="1" lexsn="2:31:00::">factors</wf>
<wf cmd="done" pos="RB" lemma="completely" wnsn="2" lexsn="4:02:03::">completely</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<wf cmd="ignore" pos="IN">into</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="product" wnsn="3" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280957.006">product</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::">degree</wf>
<wf cmd="done" pos="NN" lemma="1" wnsn="1" lexsn="1:23:00::">1</wf>
<punc>,</punc>
<wf cmd="ignore" pos="EX">there</wf>
<wf cmd="done" pos="VB" ot="notag">may</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.004">not</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="4" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">be</wf>
<wf cmd="done" pos="JJ" lemma="enough" wnsn="1" lexsn="5:00:00:sufficient:00">enough</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vectors</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="span" wnsn="1" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292743.007">span</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>;</punc>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="7" lexsn="2:42:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="clearly" wnsn="1" lexsn="4:02:00::">clearly</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="deficiency" wnsn="1" lexsn="1:26:00::">deficiency</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
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<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="JJ" lemma="second" wnsn="0" lexsn="5:00:00:intermediate:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1288373.001">second</wf>
<wf cmd="done" pos="NN" lemma="situation" wnsn="1" lexsn="1:26:00::">situation</wf>
<wf cmd="done" pos="VBZ" ot="notag">is</wf>
<wf cmd="done" pos="VB" lemma="illustrate" wnsn="1" lexsn="2:32:00::">illustrated</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">on</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<wf cmd="ignore" pos="DT">any</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1246904.016">field</wf>
<punc>)</punc>
<wf cmd="done" pos="VB" lemma="represent" wnsn="1" lexsn="2:42:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1284352.008">represented</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="standard" wnsn="1" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1293805.016">standard</wf>
<wf cmd="done" pos="NN" lemma="basis" wnsn="2" lexsn="1:09:00::">basis</wf>
<wf cmd="ignore" pos="IN">by</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
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<s snum="8">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="DT">A</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="DT" ot="notag">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="plainly" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1278534.001">plainly</wf>
<wf cmd="done" pos="RB" lemma="also" wnsn="1" lexsn="4:02:00::">also</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="DT">A</wf>
<punc>(</punc>
<wf cmd="ignore" pos="CC">or</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>)</punc>
<punc>.</punc>
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<s snum="9">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.001">not</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<punc>.</punc>
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<s snum="10">
<wf cmd="done" pos="CD" ot="notag">One</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::">sees</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="happen" wnsn="2" lexsn="2:30:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253509.001">happens</wf>
<wf cmd="ignore" pos="IN">because</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="2" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">has</wf>
<wf cmd="done" pos="NN" lemma="dimension" wnsn="1" lexsn="1:07:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1240404.002">dimension</wf>
<wf cmd="done" pos="NN" lemma="1" wnsn="1" lexsn="1:23:00::">1</wf>
<wf cmd="done" pos="RB" lemma="only" wnsn="2" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273443.001">only</wf>
<punc>.</punc>
</s>
<s snum="11">
<wf cmd="done" pos="RB" lemma="on_the_other_hand" wnsn="1" lexsn="4:02:00::">On_the_other_hand</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="together" wnsn="0" lexsn="4:02:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1299183.001">together</wf>
<wf cmd="done" pos="VB" lemma="span" wnsn="1" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292743.007">span</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="former" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1248537.003">former</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">being</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<wf cmd="done" pos="VB" lemma="span" wnsn="1" lexsn="2:42:00::">spanned</wf>
<wf cmd="ignore" pos="IN">by</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="latter" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1262393.004">latter</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<wf cmd="done" pos="VB" lemma="span" wnsn="1" lexsn="2:42:00::">spanned</wf>
<wf cmd="ignore" pos="IN">by</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="2">
<s snum="12">
<wf cmd="ignore" pos="DT">This</wf>
<wf cmd="done" pos="VB" ot="notag">will</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="7" lexsn="2:42:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">be</wf>
<wf cmd="done" pos="RB" lemma="more_or_less" wnsn="1" lexsn="4:02:00::">more_or_less</wf>
<wf cmd="ignore" pos="PRP$">our</wf>
<wf cmd="done" pos="JJ" lemma="general" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1250413.001">general</wf>
<wf cmd="done" pos="NN" lemma="method" wnsn="1" lexsn="1:09:00::">method</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="second" wnsn="1" lexsn="5:00:00:ordinal:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1288373.001">second</wf>
<wf cmd="done" pos="NN" lemma="problem" wnsn="2" lexsn="1:10:00::">problem</wf>
<punc>.</punc>
</s>
<s snum="13">
<wf cmd="ignore" pos="IN">If</wf>
<punc>(</punc>
<wf cmd="done" pos="VB" lemma="remember" wnsn="2" lexsn="2:31:01::">remember</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="NN" lemma="assumption" wnsn="2" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1223417.001">assumption</wf>
<punc>)</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="decompose" wnsn="1" lexsn="2:30:00::">decomposes</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="distinct" wnsn="1" lexsn="5:00:00:different:00">distinct</wf>
<wf cmd="done" pos="NN" lemma="element" wnsn="2" lexsn="1:06:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1243604.001">elements</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::">then</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="ignore" pos="MD">shall</wf>
<wf cmd="done" pos="VB" lemma="show" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1290298.015">show</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" ot="notag">direct_sum</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_spaces</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="14">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="special" wnsn="1" lexsn="5:00:01:specific:00">special</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="1" lexsn="1:11:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">case</wf>
<wf cmd="done" pos="RB" ot="notag">of</wf>
<wf cmd="done" pos="RB" ot="notag">this</wf>
<wf cmd="done" pos="RB" ot="notag">in</wf>
<wf cmd="done" pos="RB" ot="notag">which</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<wf cmd="done" pos="NN" ot="notag">i</wf>
<punc>.</punc>
</s>
<s snum="15">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">theorem</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="prove" wnsn="4" lexsn="2:31:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281368.001">prove</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="more" wnsn="1" lexsn="3:00:01::">more</wf>
<wf cmd="done" pos="JJ" lemma="general" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1250413.001">general</wf>
<wf cmd="ignore" pos="IN">than</wf>
<wf cmd="ignore" pos="WP">what</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VBP" ot="notag">have</wf>
<wf cmd="done" pos="VB" lemma="describe" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239683.001">described</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">since</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="work" wnsn="4" lexsn="2:35:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1307050.020">works</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="primary" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280681.002">primary</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">whether</wf>
<wf cmd="ignore" pos="CC">or</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::">not</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="prime" wnsn="1" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280698.012">primes</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="enter" wnsn="1" lexsn="2:38:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1244298.001">enter</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="all" wnsn="1" lexsn="3:00:00::">all</wf>
<wf cmd="done" pos="JJ" ot="notag">of</wf>
<wf cmd="done" pos="JJ" lemma="first" wnsn="2" lexsn="5:00:00:ordinal:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247284.001">first</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::">degree</wf>
<punc>.</punc>
</s>
<s snum="16">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="NN" lemma="reader" wnsn="1" lexsn="1:18:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1283252.001">reader</wf>
<wf cmd="ignore" pos="MD">will</wf>
<wf cmd="done" pos="VB" lemma="find" wnsn="5" lexsn="2:31:10::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247107.008">find</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="JJ" lemma="helpful" wnsn="1" lexsn="3:00:00::">helpful</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="think_of" wnsn="1" lexsn="2:31:00::">think_of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="special" wnsn="1" lexsn="5:00:01:specific:00">special</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="1" lexsn="1:11:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">case</wf>
<wf cmd="ignore" pos="WRB">when</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="prime" wnsn="1" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280698.012">primes</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::">degree</wf>
<wf cmd="done" pos="NN" lemma="1" wnsn="1" lexsn="1:23:00::">1</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="even" wnsn="1" lexsn="4:02:00::">even</wf>
<wf cmd="done" pos="JJ" lemma="more" wnsn="1" lexsn="3:00:01::">more</wf>
<wf cmd="done" pos="RB" lemma="particularly" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1275820.001">particularly</wf>
<punc>,</punc>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="think_of" wnsn="1" lexsn="2:31:00::">think_of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">10</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="special" wnsn="1" lexsn="5:00:01:specific:00">special</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="1" lexsn="1:11:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">case</wf>
<wf cmd="done" pos="RB" ot="notag">of</wf>
<wf cmd="done" pos="RB" ot="notag">this</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">theorem</wf>
<punc>.</punc>
</s>
</p>
<p pnum="3">
<s snum="17">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="finite" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247165.001">finite</wf>
<wf cmd="done" pos="JJ" lemma="dimensional" wnsn="1" lexsn="3:01:00::">dimensional</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">field</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<punc>.</punc>
</s>
<s snum="18">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<wf cmd="ignore" pos="DT">the</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="distinct" wnsn="1" lexsn="5:00:00:different:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241044.001">distinct</wf>
<wf cmd="done" pos="JJ" lemma="irreducible" wnsn="1" lexsn="3:00:00::">irreducible</wf>
<wf cmd="done" pos="NN" lemma="monic_polynomial" wnsn="1" lexsn="1:09:00::">monic_polynomials</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="positive" wnsn="1" lexsn="3:00:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1279713.014">positive</wf>
<wf cmd="done" pos="NN" lemma="integer" wnsn="1" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1258454.001">integers</wf>
<punc>.</punc>
</s>
<s snum="19">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="20">
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">Then</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">b</wf>
<punc>)</punc>
<wf cmd="ignore" pos="DT">each</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="invariant" wnsn="1" lexsn="5:00:02:invariable:00">invariant</wf>
<wf cmd="ignore" pos="IN">under</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">c</wf>
<punc>)</punc>
<wf cmd="ignore" pos="IN">if</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="induce" wnsn="1" lexsn="2:36:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1257823.002">induced</wf>
<wf cmd="ignore" pos="IN">on</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="4">
<s snum="21">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="NN" lemma="idea" wnsn="1" lexsn="1:09:00::">idea</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">this</wf>
<punc>.</punc>
</s>
<s snum="22">
<wf cmd="ignore" pos="IN">If</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="direct_sum" wnsn="1" lexsn="1:14:00::">direct_sum</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="valid" wnsn="1" lexsn="3:00:00::">valid</wf>
<punc>,</punc>
<wf cmd="ignore" pos="WRB">how</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="get" wnsn="1" lexsn="2:40:00::">get</wf>
<wf cmd="done" pos="NN" lemma="hold" wnsn="2" lexsn="1:09:00::">hold</wf>
<wf cmd="done" pos="VB" ot="notag">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="projection" wnsn="1" lexsn="1:09:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281041.001">projections</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="associate" wnsn="1" lexsn="2:31:00::">associated</wf>
<wf cmd="done" pos="VB" ot="notag">with</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<punc>?</punc>
</s>
<s snum="23">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="NN" lemma="projection" wnsn="1" lexsn="1:09:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281041.001">projection</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" ot="notag">will</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="identity" wnsn="3" lexsn="1:24:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1256928.007">identity</wf>
<wf cmd="ignore" pos="IN">on</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" lemma="zero" wnsn="1" lexsn="1:23:02::">zero</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="other" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1274198.001">other</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="24">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VB" ot="notag">shall</wf>
<wf cmd="done" pos="VB" lemma="find" wnsn="7" lexsn="2:40:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247107.008">find</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="identity" wnsn="3" lexsn="1:24:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1256928.007">identity</wf>
<wf cmd="ignore" pos="IN">on</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="NN" lemma="zero" wnsn="1" lexsn="1:23:02::">zero</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="other" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1274198.001">other</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<wf cmd="done" pos="RB" ot="notag">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="FW">etc.</wf>
<punc>.</punc>
</s>
</p>
<p pnum="5">
<s snum="25">
<wf cmd="ignore" pos="IN">For</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<wf cmd="done" pos="NN" ot="notag">i</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">let</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="26">
<wf cmd="ignore" pos="IN">Since</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="distinct" wnsn="1" lexsn="5:00:00:different:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241044.001">distinct</wf>
<wf cmd="done" pos="JJ" lemma="prime" wnsn="2" lexsn="5:00:01:first:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280698.006">prime</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="RB" lemma="relatively" wnsn="1" lexsn="4:02:00::">relatively</wf>
<wf cmd="done" pos="JJ" lemma="prime" wnsn="2" lexsn="5:00:01:first:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280698.006">prime</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">8</wf>
<punc>,</punc>
<wf cmd="done" pos="NN" lemma="chapter" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1232115.001">Chapter</wf>
<wf cmd="done" pos="CD" ot="notag">4</wf>
<punc>)</punc>
<punc>.</punc>
</s>
<s snum="27">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="ignore" pos="EX">there</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="4" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">are</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="28">
<wf cmd="done" pos="VB" lemma="note" wnsn="2" lexsn="2:39:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272172.013">Note</wf>
<wf cmd="done" pos="RB" lemma="also" wnsn="1" lexsn="4:02:00::">also</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="IN">if</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="divisible_by" wnsn="0" lexsn="5:00:00:divisible:00">divisible_by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">because</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="contain" wnsn="1" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1235596.001">contains</wf>
<wf cmd="ignore" pos="DT">each</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="IN" ot="notag">as</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="factor" wnsn="3" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245963.003">factor</wf>
<punc>.</punc>
</s>
<s snum="29">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="ignore" pos="MD">shall</wf>
<wf cmd="done" pos="VB" lemma="show" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1290298.015">show</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="behave" wnsn="3" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225722.001">behave</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="manner" wnsn="1" lexsn="1:07:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1265852.001">manner</wf>
<wf cmd="done" pos="VB" lemma="describe" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239683.001">described</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="first" wnsn="2" lexsn="5:00:00:ordinal:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247284.001">first</wf>
<wf cmd="done" pos="NN" lemma="paragraph" wnsn="1" lexsn="1:10:00::">paragraph</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<punc>.</punc>
</s>
</p>
<p pnum="6">
<s snum="30">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="31">
<wf cmd="ignore" pos="IN">Since</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="done" pos="VB" lemma="divide" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241148.007">divides</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">for</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="32">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="ignore" pos="DT">the</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="NN" lemma="projection" wnsn="1" lexsn="1:09:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281041.001">projections</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="correspond" wnsn="2" lexsn="2:42:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1236222.001">correspond</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="done" pos="JJ" lemma="some" wnsn="0" lexsn="5:00:00:unspecified:00">some</wf>
<wf cmd="done" pos="NN" ot="notag">direct_sum</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>.</punc>
</s>
<s snum="33">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VB" lemma="wish" wnsn="0" lexsn="2:31:00::">wish</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="done" pos="VB" lemma="show" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1290298.015">show</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="exactly" wnsn="1" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245352.001">exactly</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="34">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="VBZ" ot="notag">is</wf>
<wf cmd="done" pos="JJ" lemma="clear" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233608.001">clear</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="3" lexsn="2:42:05::">is</wf>
<wf cmd="ignore" pos="IN">in</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="done" pos="NN" ot="notag">|a</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="3" lexsn="2:42:05::">is</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">because</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="divisible_by" wnsn="0" lexsn="5:00:00:divisible:00">divisible_by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<punc>.</punc>
</s>
<s snum="35">
<wf cmd="done" pos="RB" lemma="conversely" wnsn="1" lexsn="4:02:00::">Conversely</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="suppose" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295947.001">suppose</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">|a</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="3" lexsn="2:42:05::">is</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="36">
<wf cmd="ignore" pos="IN">If</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="divisible_by" wnsn="0" lexsn="5:00:00:divisible:00">divisible_by</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="37">
<wf cmd="done" pos="RB" ot="notag">But</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="immediate" wnsn="1" lexsn="5:00:00:present(a):01">immediate</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">|a</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="3" lexsn="2:42:05::">is</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="38">
<wf cmd="ignore" pos="DT">This</wf>
<wf cmd="done" pos="VB" lemma="complete" wnsn="1" lexsn="2:30:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234960.007">completes</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="statement" wnsn="2" lexsn="1:10:02::">statement</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<punc>.</punc>
</s>
</p>
<p pnum="7">
<s snum="39">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="certainly" wnsn="1" lexsn="4:02:00::">certainly</wf>
<wf cmd="done" pos="JJ" lemma="clear" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233608.001">clear</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspaces</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="invariant" wnsn="1" lexsn="5:00:02:invariable:00">invariant</wf>
<wf cmd="ignore" pos="IN">under</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
<s snum="40">
<wf cmd="ignore" pos="IN">If</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="induce" wnsn="1" lexsn="2:36:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1257823.002">induced</wf>
<wf cmd="ignore" pos="IN">on</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="RB" lemma="evidently" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245318.001">evidently</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">because</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="done" pos="NN" lemma="definition" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239042.001">definition</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="NN" lemma="0" wnsn="1" lexsn="1:23:00::">0</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="41">
<wf cmd="ignore" pos="DT">This</wf>
<wf cmd="done" pos="VB" lemma="show" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1290298.015">shows</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="divide" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241148.007">divides</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="42">
<wf cmd="done" pos="RB" lemma="conversely" wnsn="1" lexsn="4:02:00::">Conversely</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">let</wf>
<wf cmd="done" pos="NN" ot="notag">g</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">any</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="43">
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">Then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="44">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="divisible_by" wnsn="0" lexsn="5:00:00:divisible:00">divisible_by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="divide" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241148.007">divides</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="45">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="VBZ" ot="notag">is</wf>
<wf cmd="done" pos="RB" lemma="easily" wnsn="1" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1242860.001">easily</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1428150.010">seen</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="divide" wnsn="2" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241148.007">divides</wf>
<wf cmd="done" pos="NN" ot="notag">g</wf>
<punc>.</punc>
</s>
<s snum="46">
<wf cmd="done" pos="RB" lemma="hence" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1254439.001">Hence</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="8">
<s snum="47">
<wf cmd="ignore" pos="IN">If</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="projection" wnsn="1" lexsn="1:09:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281041.001">projections</wf>
<wf cmd="done" pos="VB" lemma="associate" wnsn="1" lexsn="2:31:00::">associated</wf>
<wf cmd="done" pos="VB" ot="notag">with</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="primary" wnsn="1" lexsn="3:00:00::">primary</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="accordingly" wnsn="1" lexsn="4:02:00::">accordingly</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="done" pos="NNP" ot="notag">U</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commutes</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::">then</wf>
<wf cmd="done" pos="NNP" ot="notag">U</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commutes</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="DT" ot="notag">each</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="invariant" wnsn="1" lexsn="5:00:02:invariable:00">invariant</wf>
<wf cmd="ignore" pos="IN">under</wf>
<wf cmd="done" pos="NNP" ot="notag">U</wf>
<punc>.</punc>
</s>
</p>
<p pnum="9">
<s snum="48">
<wf cmd="ignore" pos="IN">In</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="notation" wnsn="1" lexsn="1:10:00::">notation</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">12</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::">let</wf>
<wf cmd="ignore" pos="PRP">us</wf>
<wf cmd="done" pos="VB" lemma="take_a_look" wnsn="1" lexsn="2:39:00::">take_a_look</wf>
<wf cmd="done" pos="VB" ot="notag">at</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="special" wnsn="1" lexsn="5:00:01:specific:00">special</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="1" lexsn="1:11:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">case</wf>
<wf cmd="done" pos="RB" ot="notag">in</wf>
<wf cmd="done" pos="RB" ot="notag">which</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="product" wnsn="3" lexsn="1:09:00::">product</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="JJ" lemma="first" wnsn="2" lexsn="5:00:00:ordinal:00">first</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239101.003">degree</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="case" wnsn="1" lexsn="1:11:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1231100.001">case</wf>
<wf cmd="done" pos="RB" ot="notag">in</wf>
<wf cmd="done" pos="RB" ot="notag">which</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="form" wnsn="2" lexsn="1:09:01::">form</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="49">
<wf cmd="done" pos="RB" lemma="now" wnsn="4" lexsn="4:02:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.001">Now</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="50">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::">Let</wf>
<wf cmd="ignore" pos="PRP">us</wf>
<wf cmd="done" pos="VB" lemma="put" wnsn="2" lexsn="2:35:05::">put</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="51">
<wf cmd="ignore" pos="IN">By</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">10</wf>
<punc>,</punc>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="ignore" pos="MD">shall</wf>
<wf cmd="done" pos="VB" lemma="call" wnsn="1" lexsn="2:32:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1230162.015">call</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="part" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1275784.001">part</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
<s snum="52">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::">Let</wf>
<wf cmd="ignore" pos="PRP">us</wf>
<wf cmd="done" pos="VB" lemma="look_at" wnsn="1" lexsn="2:31:00::">look_at</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="53">
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">Now</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="54">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="NN" lemma="reader" wnsn="1" lexsn="1:18:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1283252.001">reader</wf>
<wf cmd="ignore" pos="MD">should</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="done" rdf="familiar_with" pos="JJ" lemma="familiar_with" wnsn="0" lexsn="5:00:00:familiar:00">familiar</wf>
<wf cmd="done" pos="RB" lemma="enough" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1244241.003">enough</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="NN" lemma="projection" wnsn="1" lexsn="1:09:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281041.001">projections</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">now</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<wf cmd="done" pos="RB" ot="notag">that</wf>
<wf cmd="ignore" pos="PRP">he</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1428150.010">sees</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="in_general" wnsn="1" lexsn="4:02:00::">in_general</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="55">
<wf cmd="ignore" pos="WRB">When</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="JJ" lemma="each" wnsn="1" lexsn="5:00:00:all(a):00">each</wf>
<wf cmd="done" pos="NN" ot="notag">i</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" ot="notag">shall</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">because</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="MD">will</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="done" pos="NN" lemma="0" wnsn="1" lexsn="1:23:00::">0</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="range" wnsn="0" lexsn="1:15:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1282974.001">range</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="10">
<s snum="56">
<wf cmd="done" pos="VB" ot="notag">Let</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>.</punc>
</s>
<s snum="57">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VB" lemma="say" wnsn="2" lexsn="2:32:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1287486.008">say</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="EX">there</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="4" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">is</wf>
<wf cmd="done" pos="DT" ot="notag">some</wf>
<wf cmd="done" pos="JJ" lemma="positive" wnsn="1" lexsn="3:00:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1279713.014">positive</wf>
<wf cmd="done" pos="NN" lemma="integer" wnsn="1" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1258454.001">integer</wf>
<wf cmd="done" pos="NN" ot="notag">r</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="11">
<s snum="58">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="finite" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247165.001">finite</wf>
<wf cmd="done" pos="JJ" lemma="dimensional" wnsn="2" lexsn="5:00:00:multidimensional:00">dimensional</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">field</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<punc>.</punc>
</s>
<s snum="59">
<wf cmd="done" pos="VB" lemma="suppose" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295947.001">Suppose</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="decompose" wnsn="1" lexsn="2:30:00::">decomposes</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<wf cmd="ignore" pos="IN">into</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="product" wnsn="3" lexsn="1:09:00::">product</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="JJ" lemma="linear" wnsn="1" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263596.001">linear</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<punc>.</punc>
</s>
<s snum="60">
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">Then</wf>
<wf cmd="ignore" pos="EX">there</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="4" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">b</wf>
<punc>)</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="61">
<wf cmd="ignore" pos="DT">The</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="VBP" ot="notag">are</wf>
<wf cmd="done" pos="RB" lemma="uniquely" wnsn="1" lexsn="4:02:00::">uniquely</wf>
<wf cmd="done" pos="VB" lemma="determine" wnsn="3" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239831.003">determined</wf>
<wf cmd="ignore" pos="IN">by</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">b</wf>
<punc>)</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="DT" ot="notag">each</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="PRP">them</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
</p>
<p pnum="12">
<s snum="62">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VBP" ot="notag">have</wf>
<wf cmd="done" pos="RB" lemma="just" wnsn="3" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1260351.006">just</wf>
<wf cmd="done" pos="VB" lemma="observe" wnsn="1" lexsn="2:39:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272707.001">observed</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="done" pos="VB" lemma="write" wnsn="1" lexsn="2:36:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1307274.001">write</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="RB" lemma="not_only" wnsn="0" lexsn="4:02:00::">not_only</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="CC">but</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomials</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
<s snum="63">
<wf cmd="done" pos="RB" lemma="now" wnsn="4" lexsn="4:02:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.001">Now</wf>
<wf cmd="done" pos="VB" lemma="suppose" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295947.001">suppose</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="RB" lemma="also" wnsn="1" lexsn="4:02:00::">also</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<punc>,</punc>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="64">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="ignore" pos="MD">shall</wf>
<wf cmd="done" pos="VB" lemma="prove" wnsn="4" lexsn="2:31:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281368.001">prove</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="13">
<s snum="65">
<wf cmd="ignore" pos="IN">Since</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>'</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="done" pos="DT" ot="notag">another</wf>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1428150.010">see</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>'</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
<s snum="66">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>'</punc>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="ignore" pos="DT">any</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>;</punc>
<wf cmd="done" pos="RB" lemma="hence" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1254439.001">hence</wf>
<wf cmd="ignore" pos="PRP">they</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>.</punc>
</s>
<s snum="67">
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">Now</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">or</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="JJ" lemma="all" wnsn="1" lexsn="3:00:00::">all</wf>
<wf cmd="done" pos="JJ" lemma="four" wnsn="1" lexsn="5:00:00:cardinal:00">four</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">these</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operators</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="done" pos="DT" ot="notag">another</wf>
<punc>.</punc>
</s>
<s snum="68">
<wf cmd="ignore" pos="IN">Since</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="DT" ot="notag">both</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="PRP">they</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">they</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="RB" lemma="simultaneously" wnsn="1" lexsn="4:02:00::">simultaneously</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<punc>.</punc>
</s>
<s snum="69">
<wf cmd="ignore" pos="IN">Since</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="DT" ot="notag">both</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="PRP">they</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<punc>;</punc>
<wf cmd="done" pos="RB" ot="notag">for</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="use" wnsn="1" lexsn="2:34:01::">using</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="fact" wnsn="2" lexsn="1:10:01::">fact</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<punc>'</punc>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" ot="notag">so</wf>
<wf cmd="ignore" pos="WRB">when</wf>
<wf cmd="done" pos="NN" ot="notag">r</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="sufficiently" wnsn="1" lexsn="4:02:00::">sufficiently</wf>
<wf cmd="done" pos="JJ" lemma="large" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1262244.001">large</wf>
<wf cmd="ignore" pos="DT">every</wf>
<wf cmd="done" pos="NN" lemma="term" wnsn="4" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1297843.013">term</wf>
<wf cmd="done" pos="RB" ot="notag">in</wf>
<wf cmd="done" pos="RB" ot="notag">this</wf>
<wf cmd="done" pos="NN" lemma="expression" wnsn="6" lexsn="1:10:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245687.005">expression</wf>
<wf cmd="ignore" pos="IN">for</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="MD">will</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="done" pos="NN" lemma="0" wnsn="1" lexsn="1:23:00::">0</wf>
<punc>.</punc>
</s>
<s snum="70">
<punc>(</punc>
<wf cmd="done" pos="RB" lemma="actually" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1219820.002">Actually</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="JJ" ot="notag">n-dimensional</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::">space</wf>
<wf cmd="ignore" pos="MD">must</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<wf cmd="ignore" pos="PRP$">its</wf>
<wf cmd="done" pos="NN" ot="notag">nth</wf>
<wf cmd="done" pos="NN" lemma="power" wnsn="6" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280001.022">power</wf>
<wf cmd="done" pos="NN" lemma="0" wnsn="1" lexsn="1:23:00::">0</wf>
<punc>;</punc>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="take" wnsn="13" lexsn="2:31:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1296844.001">take</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="above" wnsn="1" lexsn="4:02:00::">above</wf>
<punc>,</punc>
<wf cmd="ignore" pos="WDT">that</wf>
<wf cmd="ignore" pos="MD">will</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="done" pos="JJ" lemma="large" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1262244.001">large</wf>
<wf cmd="done" pos="JJ" lemma="enough" wnsn="1" lexsn="5:00:00:sufficient:00">enough</wf>
<punc>.</punc>
</s>
<s snum="71">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="VB" lemma="follow" wnsn="3" lexsn="2:42:05::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1248194.006">follows</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="large" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1262244.001">large</wf>
<wf cmd="done" pos="JJ" lemma="enough" wnsn="1" lexsn="5:00:00:sufficient:00">enough</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">but</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.001">not</wf>
<wf cmd="done" pos="JJ" lemma="obvious" wnsn="1" lexsn="3:00:00::">obvious</wf>
<wf cmd="ignore" pos="IN">from</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="above" wnsn="1" lexsn="5:00:00:preceding(a):00">above</wf>
<wf cmd="done" pos="NN" lemma="expression" wnsn="6" lexsn="1:10:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245687.005">expression</wf>
<punc>.</punc>
<punc>)</punc>
</s>
<s snum="72">
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">Now</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="also" wnsn="1" lexsn="4:02:00::">also</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<punc>.</punc>
</s>
<s snum="73">
<wf cmd="done" pos="JJ" lemma="such" wnsn="0" lexsn="5:00:01:specified:00">Such</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="obviously" wnsn="1" lexsn="4:02:00::">obviously</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="zero" wnsn="1" lexsn="1:23:02::">zero</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<punc>;</punc>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="IN">since</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="form" wnsn="2" lexsn="1:09:01::">form</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="JJ" lemma="some" wnsn="0" lexsn="5:00:00:unspecified:00">some</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>;</punc>
<wf cmd="done" pos="RB" ot="notag">but</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="ignore" pos="IN">since</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="MD">cannot</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">have</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="repeated" wnsn="1" lexsn="5:00:00:continual:00">repeated</wf>
<wf cmd="done" pos="NN" lemma="root" wnsn="4" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1285766.013">root</wf>
<punc>;</punc>
<wf cmd="done" pos="RB" lemma="hence" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1254439.001">hence</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="simply" wnsn="1" lexsn="4:02:00::">simply</wf>
<wf cmd="done" pos="NN" ot="notag">x</wf>
<punc>,</punc>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="say" wnsn="2" lexsn="2:32:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1287486.001">says</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="6" lexsn="2:42:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="NN" lemma="0" wnsn="1" lexsn="1:23:00::">0</wf>
<punc>.</punc>
</s>
<s snum="74">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1428150.010">see</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="14">
<s snum="75">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="finite" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247165.001">finite</wf>
<wf cmd="done" pos="JJ" lemma="dimensional" wnsn="2" lexsn="5:00:00:multidimensional:00">dimensional</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="RB" lemma="algebraically" wnsn="1" lexsn="4:02:00::">algebraically</wf>
<wf cmd="done" pos="JJ" lemma="closed" wnsn="2" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233818.001">closed</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">field</wf>
<wf cmd="done" pos="NN" ot="notag">F</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="e.g." wnsn="1" lexsn="4:02:00::">e.g.</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">field</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="complex_number" wnsn="1" lexsn="1:23:00::">complex_numbers</wf>
<punc>.</punc>
</s>
<s snum="76">
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">Then</wf>
<wf cmd="done" pos="JJ" lemma="every" wnsn="2" lexsn="5:00:02:all(a):00">every</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="done" pos="VB" ot="notag">be</wf>
<wf cmd="done" pos="VB" lemma="write" wnsn="1" lexsn="2:36:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1307274.001">written</wf>
<wf cmd="ignore" pos="IN">as</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="sum" wnsn="3" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295634.002">sum</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="diagonalizable" wnsn="1" lexsn="3:01:00::">diagonalizable</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="commute" wnsn="1" lexsn="2:30:04::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234867.005">commute</wf>
<punc>.</punc>
</s>
<s snum="77">
<wf cmd="ignore" pos="DT">These</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operators</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">N</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="unique" wnsn="0" lexsn="5:00:00:single(a):00">unique</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">each</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
</p>
<p pnum="15">
<s snum="78">
<wf cmd="ignore" pos="IN">From</wf>
<wf cmd="ignore" pos="DT">these</wf>
<wf cmd="done" pos="NN" lemma="result" wnsn="1" lexsn="1:19:00::">results</wf>
<punc>,</punc>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="done" pos="VB" lemma="see" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1428150.010">sees</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="study" wnsn="2;1" lexsn="1:09:03::;1:04:00::">study</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operators</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">spaces</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="RB" lemma="algebraically" wnsn="1" lexsn="4:02:00::">algebraically</wf>
<wf cmd="done" pos="JJ" lemma="closed" wnsn="2" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233818.001">closed</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">field</wf>
<wf cmd="done" pos="VBZ" ot="notag">is</wf>
<wf cmd="done" pos="RB" lemma="essentially" wnsn="1" lexsn="4:02:01::">essentially</wf>
<wf cmd="done" pos="VB" lemma="reduce" wnsn="6" lexsn="2:30:11::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1283720.012">reduced</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="study" wnsn="2;1" lexsn="1:09:03::;1:04:00::">study</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="JJ" lemma="nilpotent" wnsn="1" lexsn="3:01:00::">nilpotent</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operators</wf>
<punc>.</punc>
</s>
<s snum="79">
<wf cmd="ignore" pos="IN">For</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">spaces</wf>
<wf cmd="ignore" pos="IN">over</wf>
<wf cmd="done" pos="JJ" lemma="non" wnsn="0" lexsn="5:00:00:negative:02">non</wf>
<wf cmd="done" pos="RB" lemma="algebraically" wnsn="1" lexsn="4:02:00::">algebraically</wf>
<wf cmd="done" pos="JJ" lemma="closed" wnsn="2" lexsn="3:00:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1233818.001">closed</wf>
<wf cmd="done" pos="NN" lemma="field" wnsn="10" lexsn="1:14:00::">fields</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="RB" lemma="still" wnsn="1" lexsn="4:02:01::">still</wf>
<wf cmd="done" pos="VB" ot="notag">need</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="done" pos="VB" lemma="find" wnsn="7" lexsn="2:40:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247107.008">find</wf>
<wf cmd="done" pos="JJ" lemma="some" wnsn="0" lexsn="5:00:00:unspecified:00">some</wf>
<wf cmd="done" pos="NN" lemma="substitute" wnsn="1" lexsn="1:09:00::">substitute</wf>
<wf cmd="done" pos="VB" ot="notag">for</wf>
<wf cmd="done" pos="JJ" lemma="characteristic" wnsn="1" lexsn="3:00:00::">characteristic</wf>
<wf cmd="done" pos="NN" lemma="value" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303448.007">values</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vectors</wf>
<punc>.</punc>
</s>
<s snum="80">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="RB" lemma="very" wnsn="1" lexsn="4:02:00::">very</wf>
<wf cmd="done" pos="JJ" lemma="interesting" wnsn="1" lexsn="3:00:00::">interesting</wf>
<wf cmd="done" pos="NN" lemma="fact" wnsn="2" lexsn="1:10:01::">fact</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">these</wf>
<wf cmd="done" pos="JJ" lemma="two" wnsn="1" lexsn="5:00:00:cardinal:00">two</wf>
<wf cmd="done" pos="NN" lemma="problem" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1280894.001">problems</wf>
<wf cmd="ignore" pos="MD">can</wf>
<wf cmd="done" pos="VB" ot="notag">be</wf>
<wf cmd="done" pos="VB" lemma="handle" wnsn="3" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253407.004">handled</wf>
<wf cmd="done" pos="RB" lemma="simultaneously" wnsn="1" lexsn="4:02:00::">simultaneously</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="7" lexsn="2:42:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="WP">what</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="ignore" pos="MD">shall</wf>
<wf cmd="done" pos="VB" ot="notag">do</wf>
<wf cmd="done" pos="VB" ot="notag">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="next" wnsn="1" lexsn="5:00:00:succeeding(a):00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1271334.004">next</wf>
<wf cmd="done" pos="NN" lemma="chapter" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1232115.001">chapter</wf>
<punc>.</punc>
</s>
</p>
<p pnum="16">
<s snum="81">
<wf cmd="ignore" pos="IN">In</wf>
<wf cmd="done" pos="VB" lemma="conclude" wnsn="2" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1235148.001">concluding</wf>
<wf cmd="ignore" pos="DT">this</wf>
<wf cmd="done" pos="NN" lemma="section" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1288470.001">section</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" ot="notag">should_like</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="give" wnsn="4" lexsn="2:40:07::">give</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="NN" lemma="example" wnsn="2" lexsn="1:09:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1245362.001">example</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="illustrate" wnsn="1" lexsn="2:32:00::">illustrates</wf>
<wf cmd="done" pos="DT" ot="notag">some</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="idea" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1256909.001">ideas</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="primary" wnsn="1" lexsn="3:00:00::">primary</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">theorem</wf>
<punc>.</punc>
</s>
<s snum="82">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="VBP" ot="notag">have</wf>
<wf cmd="done" pos="VB" lemma="choose" wnsn="1" lexsn="2:31:00::">chosen</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="give" wnsn="4" lexsn="2:40:07::">give</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="ignore" pos="IN">at</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="end" wnsn="5" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1244023.001">end</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="section" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1288470.001">section</wf>
<wf cmd="ignore" pos="IN">since</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="deal" wnsn="2" lexsn="2:31:10::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1238634.007">deals</wf>
<wf cmd="done" pos="VB" ot="notag">with</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equations</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">thus</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.001">not</wf>
<wf cmd="done" pos="RB" lemma="purely" wnsn="1" lexsn="4:02:00::">purely</wf>
<wf cmd="done" pos="NN" lemma="linear_algebra" wnsn="1" lexsn="1:09:00::">linear_algebra</wf>
<punc>.</punc>
</s>
</p>
<p pnum="17">
<s snum="83">
<wf cmd="ignore" pos="IN">In</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="primary" wnsn="1" lexsn="3:00:00::">primary</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">theorem</wf>
<punc>,</punc>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.001">not</wf>
<wf cmd="done" pos="JJ" lemma="necessary" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1270794.001">necessary</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="done" pos="JJ" lemma="finite" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247165.001">finite</wf>
<wf cmd="done" pos="JJ" lemma="dimensional" wnsn="1" lexsn="3:01:00::">dimensional</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">nor</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="JJ" lemma="necessary" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1270794.001">necessary</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" lemma="part" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1275784.001">parts</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">b</wf>
<punc>)</punc>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="minimal" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1268161.002">minimal</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<punc>.</punc>
</s>
<s snum="84">
<wf cmd="ignore" pos="IN">If</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">an</wf>
<wf cmd="done" pos="JJ" lemma="arbitrary" wnsn="1" lexsn="3:00:00::">arbitrary</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="EX">there</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="4" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.001">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="monic_polynomial" wnsn="1" lexsn="1:09:00::">monic_polynomial</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="NN" lemma="part" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1275784.001">parts</wf>
<punc>(</punc>
<wf cmd="ignore" pos="DT">a</wf>
<punc>)</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">b</wf>
<punc>)</punc>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">12</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="valid" wnsn="1" lexsn="3:00:00::">valid</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="done" pos="NN" ot="notag">T</wf>
<wf cmd="ignore" pos="IN">with</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="proof" wnsn="2" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1281123.001">proof</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VB" lemma="give" wnsn="4" lexsn="2:40:07::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1250951.026">gave</wf>
<punc>.</punc>
</s>
</p>
<p pnum="18">
<s snum="85">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">Let</wf>
<wf cmd="done" pos="NN" ot="notag">n</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="JJ" lemma="positive" wnsn="1" lexsn="3:00:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1279713.014">positive</wf>
<wf cmd="done" pos="NN" lemma="integer" wnsn="1" lexsn="1:23:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1258454.001">integer</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263073.003">let</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="JJ" lemma="all" wnsn="1" lexsn="3:00:00::">all</wf>
<wf cmd="done" pos="NN" ot="notag">n</wf>
<wf cmd="done" pos="NNS" ot="notag">times</wf>
<wf cmd="done" pos="RB" lemma="continuously" wnsn="1" lexsn="4:02:00::">continuously</wf>
<wf cmd="done" pos="JJ" lemma="differentiable" wnsn="1" lexsn="3:01:00::">differentiable</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="real" wnsn="1" lexsn="3:00:00::">real</wf>
<wf cmd="done" pos="NN" lemma="line" wnsn="4" lexsn="1:25:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1263591.001">line</wf>
<wf cmd="ignore" pos="WDT">which</wf>
<wf cmd="done" pos="VB" lemma="satisfy" wnsn="1" lexsn="2:42:00::">satisfy</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equation</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="ignore" pos="DT">some</wf>
<wf cmd="done" pos="JJ" lemma="fixed" wnsn="1" lexsn="5:00:00:determinate:01" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1247444.001">fixed</wf>
<wf cmd="done" pos="NN" lemma="constant" wnsn="1" lexsn="1:09:00::">constants</wf>
<punc>.</punc>
</s>
<s snum="86">
<wf cmd="ignore" pos="IN">If</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="denote" wnsn="1" lexsn="2:32:01::">denotes</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" ot="notag">n</wf>
<wf cmd="done" pos="NNS" ot="notag">times</wf>
<wf cmd="done" pos="RB" lemma="continuously" wnsn="1" lexsn="4:02:00::">continuously</wf>
<wf cmd="done" pos="JJ" lemma="differentiable" wnsn="1" lexsn="3:01:00::">differentiable</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<wf cmd="done" pos="RB" ot="notag">of</wf>
<wf cmd="done" pos="RB" ot="notag">this</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equation</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="7" lexsn="2:42:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="87">
<wf cmd="ignore" pos="IN">If</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="done" pos="VB" lemma="denote" wnsn="1" lexsn="2:32:01::">denotes</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="differentiation" wnsn="2" lexsn="1:04:01::">differentiation</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="operator" wnsn="1" lexsn="1:24:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273617.005">operator</wf>
<wf cmd="done" pos="NN" ot="notag">p</wf>
<punc>(</punc>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<punc>)</punc>
<punc>,</punc>
<wf cmd="ignore" pos="IN">because</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="simply" wnsn="1" lexsn="4:02:00::">simply</wf>
<wf cmd="done" pos="VB" lemma="say" wnsn="2" lexsn="2:32:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1287486.001">says</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="88">
<wf cmd="done" pos="VB" lemma="let" wnsn="1" lexsn="2:41:00::">Let</wf>
<wf cmd="ignore" pos="PRP">us</wf>
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">now</wf>
<wf cmd="done" pos="VB" lemma="regard" wnsn="3" lexsn="2:42:00::">regard</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="ignore" pos="IN">as</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="subspace" wnsn="1" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1295377.001">subspace</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<punc>.</punc>
</s>
<s snum="89">
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">Then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
</p>
<p pnum="19">
<s snum="90">
<wf cmd="ignore" pos="IN">If</wf>
<wf cmd="ignore" pos="PRP">we</wf>
<wf cmd="done" pos="VBP" ot="notag">are</wf>
<wf cmd="done" pos="VB" lemma="discuss" wnsn="1" lexsn="2:32:01::">discussing</wf>
<wf cmd="done" pos="JJ" lemma="differentiable" wnsn="1" lexsn="3:01:00::">differentiable</wf>
<wf cmd="done" pos="JJ" lemma="complex" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234966.003">complex</wf>
<wf cmd="done" pos="JJ" lemma="valued" wnsn="1" lexsn="5:00:00:quantitative:00">valued</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="NN" ot="notag">V</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="complex" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1234966.003">complex</wf>
<wf cmd="done" pos="NN" lemma="vector" wnsn="1" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1303671.001">vector</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">spaces</wf>
<punc>,</punc>
<wf cmd="ignore" pos="CC">and</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="MD">may</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">any</wf>
<wf cmd="done" pos="NN" lemma="complex_number" wnsn="1" lexsn="1:23:00::">complex_numbers</wf>
<punc>.</punc>
</s>
<s snum="91">
<wf cmd="ignore" pos="PRP">We</wf>
<wf cmd="done" pos="RB" lemma="now" wnsn="2" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272258.009">now</wf>
<wf cmd="done" pos="VB" lemma="write" wnsn="1" lexsn="2:36:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1307274.001">write</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="JJ" lemma="distinct" wnsn="1" lexsn="5:00:00:different:00" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1241044.001">distinct</wf>
<wf cmd="done" pos="NN" lemma="complex_number" wnsn="1" lexsn="1:23:00::">complex_numbers</wf>
<punc>.</punc>
</s>
<s snum="92">
<wf cmd="ignore" pos="IN">If</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="null_space" wnsn="1" lexsn="1:14:00::">null_space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">Theorem</wf>
<wf cmd="done" pos="CD" ot="notag">12</wf>
<wf cmd="done" pos="VB" lemma="say" wnsn="2" lexsn="2:32:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1287486.001">says</wf>
<wf cmd="ignore" pos="IN">that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="93">
<wf cmd="done" pos="RB" lemma="in_other_words" wnsn="1" lexsn="4:02:00::">In_other_words</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="satisfy" wnsn="1" lexsn="2:42:00::">satisfies</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equation</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="uniquely" wnsn="1" lexsn="4:02:00::">uniquely</wf>
<wf cmd="done" pos="JJ" lemma="expressible" wnsn="1" lexsn="3:00:00::">expressible</wf>
<wf cmd="ignore" pos="IN">in</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="form" wnsn="2" lexsn="1:09:01::">form</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="WRB">where</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="satisfy" wnsn="1" lexsn="2:42:00::">satisfies</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equation</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="94">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="study" wnsn="2" lexsn="1:09:03::">study</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="equation" wnsn="1" lexsn="1:10:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1244613.004">equation</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VBZ" ot="notag">is</wf>
<wf cmd="done" pos="VB" lemma="reduce" wnsn="6" lexsn="2:30:11::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1283720.012">reduced</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="study" wnsn="2" lexsn="1:09:03::">study</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equation</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="form" wnsn="2" lexsn="1:09:01::">form</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="95">
<wf cmd="ignore" pos="DT">This</wf>
<wf cmd="done" pos="NN" lemma="reduction" wnsn="3" lexsn="1:04:02::">reduction</wf>
<wf cmd="done" pos="VBZ" ot="notag">has</wf>
<wf cmd="done" pos="VBN" ot="notag">been</wf>
<wf cmd="done" pos="VB" lemma="accomplish" wnsn="1" lexsn="2:36:00::">accomplished</wf>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="general" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1250413.001">general</wf>
<wf cmd="done" pos="NN" lemma="method" wnsn="1" lexsn="1:09:00::">methods</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="linear_algebra" wnsn="1" lexsn="1:09:00::">linear_algebra</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<wf cmd="ignore" pos="IN">by</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="JJ" lemma="primary" wnsn="1" lexsn="3:00:00::">primary</wf>
<wf cmd="done" pos="NN" lemma="decomposition" wnsn="1" lexsn="1:09:00::">decomposition</wf>
<wf cmd="done" pos="NN" lemma="theorem" wnsn="1" lexsn="1:10:00::">theorem</wf>
<punc>.</punc>
</s>
</p>
<p pnum="20">
<s snum="96">
<wf cmd="ignore" pos="TO">To</wf>
<wf cmd="done" pos="VB" lemma="describe" wnsn="1" lexsn="2:32:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239683.001">describe</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<wf cmd="ignore" pos="TO">to</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="ignore" pos="MD">must</wf>
<wf cmd="done" pos="VB" lemma="know" wnsn="1" lexsn="2:31:01::">know</wf>
<wf cmd="done" pos="NN" ot="notag">something</wf>
<wf cmd="ignore" pos="IN">about</wf>
<wf cmd="done" pos="NN" lemma="differential_equation" wnsn="1" lexsn="1:10:00::">differential_equations</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="that_is" wnsn="0" lexsn="4:02:00::">that_is</wf>
<punc>,</punc>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="ignore" pos="MD">must</wf>
<wf cmd="done" pos="VB" lemma="know" wnsn="1" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1261363.001">know</wf>
<wf cmd="done" pos="NN" ot="notag">something</wf>
<wf cmd="ignore" pos="IN">about</wf>
<wf cmd="done" pos="NN" ot="notag">D</wf>
<wf cmd="done" pos="JJ" lemma="other_than" wnsn="0" lexsn="5:00:00:other:00">other_than</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="fact" wnsn="2" lexsn="1:10:01::">fact</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="PRP">it</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="linear_operator" wnsn="1" lexsn="1:24:00::">linear_operator</wf>
<punc>.</punc>
</s>
<s snum="97">
<wf cmd="done" pos="RB" lemma="however" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1256065.001">However</wf>
<punc>,</punc>
<wf cmd="done" pos="CD" ot="notag">one</wf>
<wf cmd="done" pos="VBZ" ot="notag">does</wf>
<wf cmd="done" pos="RB" lemma="not" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1272156.004">not</wf>
<wf cmd="done" pos="VB" ot="notag">need</wf>
<wf cmd="done" pos="VB" ot="notag">to</wf>
<wf cmd="done" pos="VB" lemma="know" wnsn="1" lexsn="2:31:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1261363.001">know</wf>
<wf cmd="done" pos="RB" lemma="very_much" wnsn="1" lexsn="4:02:01::">very_much</wf>
<punc>.</punc>
</s>
<s snum="98">
<wf cmd="ignore" pos="PRP">It</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">is</wf>
<wf cmd="done" pos="RB" lemma="very" wnsn="1" lexsn="4:02:00::">very</wf>
<wf cmd="done" pos="JJ" lemma="easy" wnsn="1" lexsn="3:00:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1242926.001">easy</wf>
<wf cmd="ignore" pos="TO">to</wf>
<wf cmd="done" pos="VB" lemma="establish" wnsn="3" lexsn="2:31:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1244934.007">establish</wf>
<wf cmd="done" pos="VB" ot="notag">by</wf>
<wf cmd="done" pos="NN" lemma="induction" wnsn="3" lexsn="1:09:00::">induction</wf>
<wf cmd="ignore" pos="IN">on</wf>
<wf cmd="done" pos="NN" ot="notag">r</wf>
<wf cmd="ignore" pos="IN">that</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="3" lexsn="2:42:05::">is</wf>
<wf cmd="ignore" pos="IN">in</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="then" wnsn="2" lexsn="4:02:02::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298166.004">then</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="RB" lemma="that_is" wnsn="0" lexsn="4:02:00::">that_is</wf>
<punc>,</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="etc." wnsn="1" lexsn="4:02:00::">etc.</wf>
<punc>.</punc>
</s>
<s snum="99">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="only" wnsn="2" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273443.001">only</wf>
<wf cmd="ignore" pos="IN">if</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="100">
<wf cmd="ignore" pos="DT">A</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">function</wf>
<wf cmd="done" pos="NN" ot="notag">g</wf>
<wf cmd="done" pos="JJ" lemma="such_that" wnsn="0" lexsn="5:00:00:specified:00">such_that</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="RB" lemma="i.e." wnsn="1" lexsn="4:02:00::">i.e.</wf>
<punc>,</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" ot="notag">must</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="2" lexsn="2:42:06::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">be</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="polynomial" wnsn="1" lexsn="1:09:00::">polynomial</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">function</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="degree" wnsn="5" lexsn="1:09:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1239101.003">degree</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="CC">or</wf>
<wf cmd="done" pos="JJ" lemma="less" wnsn="1" lexsn="3:00:00::">less</wf>
<punc>:</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="101">
<wf cmd="done" pos="RB" lemma="thus" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1298662.001">Thus</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="satisfy" wnsn="1" lexsn="2:42:00::">satisfies</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="done" pos="RB" lemma="only" wnsn="2" lexsn="4:02:01::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1273443.001">only</wf>
<wf cmd="ignore" pos="IN">if</wf>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="2" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">has</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="form" wnsn="2" lexsn="1:09:01::">form</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="102">
<wf cmd="done" pos="RB" lemma="accordingly" wnsn="1" lexsn="4:02:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1219568.002">Accordingly</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">the</wf>
<punc>'</punc>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<punc>'</punc>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="span" wnsn="1" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292743.007">span</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<wf cmd="ignore" pos="IN">of</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>.</punc>
</s>
<s snum="103">
<wf cmd="ignore" pos="IN">Since</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<wf cmd="done" pos="VB" lemma="be" wnsn="1" lexsn="2:42:03::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1225404.008">are</wf>
<wf cmd="done" pos="RB" lemma="linearly" wnsn="1" lexsn="4:02:00::">linearly</wf>
<wf cmd="done" pos="JJ" lemma="independent" wnsn="1" lexsn="3:00:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1257673.010">independent</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<wf cmd="ignore" pos="CC">and</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="exponential_function" wnsn="1" lexsn="1:24:00::">exponential_function</wf>
<wf cmd="done" pos="VB" lemma="have" wnsn="1" lexsn="2:40:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1253820.001">has</wf>
<wf cmd="done" pos="JJ" lemma="no" wnsn="1" lexsn="3:00:00::">no</wf>
<wf cmd="done" pos="NN" lemma="zero" wnsn="1" lexsn="1:23:02::">zeros</wf>
<punc>,</punc>
<wf cmd="ignore" pos="DT">these</wf>
<wf cmd="done" pos="NN" ot="notag">r</wf>
<wf cmd="done" pos="NN" lemma="function" wnsn="1" lexsn="1:24:00::">functions</wf>
<punc>*</punc>
<punc>*</punc>
<wf cmd="done" pos="NN" ot="notag">f</wf>
<punc>,</punc>
<wf cmd="done" pos="VB" lemma="form" wnsn="2" lexsn="2:42:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1248515.024">form</wf>
<wf cmd="ignore" pos="DT">a</wf>
<wf cmd="done" pos="NN" lemma="basis" wnsn="2" lexsn="1:09:00::">basis</wf>
<wf cmd="ignore" pos="IN">for</wf>
<wf cmd="ignore" pos="DT">the</wf>
<wf cmd="done" pos="NN" lemma="space" wnsn="0" lexsn="1:14:00::" oed_id="/dictionary/sense/en_us_NOAD3e_2012/m_en_us1292678.012">space</wf>
<wf cmd="ignore" pos="IN">of</wf>
<wf cmd="done" pos="NN" lemma="solution" wnsn="4" lexsn="1:14:00::">solutions</wf>
<punc>.</punc>
</s>
</p>
</context>
</contextfile>