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Процесс ортогонализации Шмидта </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="2" left="48" top="57" width="448.5" height="34.5" align-x="48" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Пусть в пространстве </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> задан произвольный базис векторов </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">2</f> <f family="Arial Cyr" charset="204" size="12">,..., </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">), которые</f> <f family="Arial Cyr" charset="204" size="12">ниже записаны в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> .</f> </text> </region> <region region-id="74" left="96" top="101.25" width="100.5" height="78.75" align-x="107.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">f</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>2</ml:real> <ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> <ml:real>1</ml:real> <ml:real>4</ml:real> <ml:real>1</ml:real> <ml:real>7</ml:real> <ml:real>3</ml:real> <ml:real>3</ml:real> 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align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Построим в этом же пространстве ортогональный базис векторов {</f> <f family="Arial Cyr" charset="204" size="12"> е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}, которые</f> <f family="Arial Cyr" charset="204" size="12">также запишем в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Процесс ортогонализации Шмидта представ-</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">лен в виде алгоритма. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">Число ортогональных векторов </f> 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push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем произвольно первый вектор:</f> </text> </region> <region region-id="90" left="288" top="300.75" width="42" height="15.75" align-x="298.5" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="91" left="354" top="300.75" width="54" height="17.25" align-x="381" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIConstant" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> 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</ml:matrix> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="55" left="66" top="708.75" width="265.5" height="13.5" align-x="66" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Проверим ортогональность полученного базиса:</f> </text> </region> <region region-id="64" left="66" top="744.75" width="33" height="15.75" align-x="82.5" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">w</ml:id> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="63" left="114" top="744.75" width="30" height="15.75" align-x="127.5" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">v</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="65" left="186" top="729.75" width="117" height="50.25" align-x="281.25" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">j</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="97" left="42" top="798" width="452.25" height="84" align-x="42" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Что и требовалось показать. Легко решает задачи такого класса при любом </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12">Однако задачу<sp count="2"/>4.70 из [24] </f> <f family="Arial Cyr" charset="204" size="12">с нулем </f> <f family="Arial Cyr" charset="204" size="12">не решает. Для этого нужен универсальный</f> <f family="Arial Cyr" charset="204" size="12">алгоритм приводимый ниже. Или всякий раз, когда </f> <f family="Arial Cyr" charset="204" size="12">появляется нулевая строка в матрице </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> с номером </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, необходимо перемещать </inlineAttr> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">в матрице </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">вектор с номером </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> q </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12"> в последнюю строку и повторять вычисления. Вычисления прекратить, вместе </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">с перебором всех нижних </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,...,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> <sp count="4"/>строк матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">f</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="120" left="18" top="906.75" width="483" height="27.75" align-x="18" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Рассмотрим процесс ортогонализации Шмидта,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">учитывающий возможную линейную зависимость векторов. </inlineAttr> </f> </text> </region> <region region-id="121" left="18" top="957" width="445.5" height="34.5" align-x="18" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В пространстве </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> задана произвольная система векторов </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12">f</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12">,..., </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">, которые </f> <f family="Arial Cyr" charset="204" size="12">ниже записаны в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> (см. </f> <f family="Arial Cyr" charset="204" size="12">задачу 4.70</f> <f family="Arial Cyr" charset="204" size="12"> из [24]).</f> </text> </region> <region region-id="122" left="24" top="1044.75" width="30" height="15.75" align-x="37.5" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">f50a74dc394bd39753bfa0ed5c3ab631</contentHash> <ml:define warning="WarnRedefinedUDScalar"> <ml:id xml:space="preserve">n</ml:id> <ml:real>4</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="16"> <element-image-map> <box left="1.5" top="0.75" width="28.5" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="123" left="78" top="1013.25" width="100.5" height="78.75" align-x="89.25" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" 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<ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> <ml:real>1</ml:real> <ml:real>4</ml:real> <ml:real>1</ml:real> <ml:real>7</ml:real> <ml:real>3</ml:real> <ml:real>3</ml:real> <ml:real>-6</ml:real> <ml:real>7</ml:real> <ml:real>-1</ml:real> <ml:real>-3</ml:real> <ml:real>0</ml:real> <ml:real>8</ml:real> </ml:matrix> </ml:define> </ml:provenance> </math> <rendering item-idref="17"> <element-image-map> <box left="1.5" top="0.75" width="99" height="77.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="124" left="222" top="1044.75" width="48.75" height="17.25" align-x="249" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">c6328b3b051a1180429b0804aa4f3493</contentHash> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="18"> <element-image-map> <box left="1.5" top="0.75" width="18.75" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="125" left="312" top="1013.25" width="62.25" height="78.75" align-x="336" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">df02640a8fdde5fdfd6e7b117dbabd17</contentHash> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">f</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>2</ml:real> <ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="19"> <element-image-map> <box left="1.5" top="27.75" width="15.75" height="18.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="126" left="18" top="1110.75" width="473.25" height="45" align-x="18" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Запишем более общий алгоритм построения в этом же пространстве ортогонального базиса векторов {</f> <f family="Arial Cyr" charset="204" size="12"> е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}, который будет учитывать возможную линейную зависимость строк матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="155" left="18" top="1170.75" width="243.75" height="13.5" align-x="18" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, выберем произвольно первый вектор:</f> </text> </region> <region region-id="156" left="288" top="1170.75" width="42" height="15.75" align-x="298.5" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" 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<hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">7680f4a2f8d24ecee199dd71e66d8dfc</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="23"> <element-image-map> <box left="1.5" top="0.75" width="32.25" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="160" left="24" top="1206" width="467.25" height="17.25" align-x="24" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Последующие векторы<sp count="2"/> </f> 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background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment 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\viewkind4\uc1\pard\qc\f0\fs18\'d2\'e8\'f2\'ee\'e2 \'ca.\'c2.\par }</center> <right>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \viewkind4\uc1\pard\qr\f0\fs18\{n\}\par }</right> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="1" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC12" local-assignment="MC12"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="cm"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="160" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="1" left="66" top="18.75" width="373.5" height="14.25" align-x="66" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> 1.4. Процесс ортогонализации Шмидта </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="2" left="48" top="57" width="448.5" height="34.5" align-x="48" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Пусть в пространстве </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> задан произвольный базис векторов </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">2</f> <f family="Arial Cyr" charset="204" size="12">,..., </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">), которые</f> <f family="Arial Cyr" charset="204" size="12">ниже записаны в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> .</f> </text> </region> <region region-id="74" left="96" top="101.25" width="100.5" height="78.75" align-x="107.25" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">f</ml:id> <ml:matrix rows="4" cols="4"> <ml:real>2</ml:real> <ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> <ml:real>1</ml:real> <ml:real>4</ml:real> <ml:real>1</ml:real> <ml:real>7</ml:real> <ml:real>3</ml:real> <ml:real>3</ml:real> <ml:real>-6</ml:real> <ml:real>7</ml:real> <ml:real>-1</ml:real> <ml:real>-3</ml:real> <ml:real>0</ml:real> <ml:real>8</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="18" left="42" top="138.75" width="30" height="15.75" align-x="55.5" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">n</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="24" left="240" top="138.75" width="48.75" height="17.25" align-x="267" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="4"/> </region> <region region-id="25" left="330" top="107.25" width="62.25" height="78.75" align-x="354" align-y="150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">f</ml:id> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>2</ml:real> <ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="5"/> </region> <region region-id="53" left="48" top="198" width="473.25" height="60" align-x="48" align-y="210" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Построим в этом же пространстве ортогональный базис векторов {</f> <f family="Arial Cyr" charset="204" size="12"> е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}, которые</f> <f family="Arial Cyr" charset="204" size="12">также запишем в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Процесс ортогонализации Шмидта представ-</inlineAttr> </f> <inlineAttr font-weight="normal"> <f family="Arial Cyr" charset="204" size="12">лен в виде алгоритма. </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12">Число ортогональных векторов </f> <f family="Arial Cyr" charset="204" size="12">q</f> <f family="Arial Cyr" charset="204" size="12"> можно задавать произ- вольно, монотонно увеличивая его до появления в матрице </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> нулевой строки.</f> </text> </region> <region region-id="96" left="60" top="270.75" width="30.75" height="15.75" align-x="74.25" align-y="282" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">q</ml:id> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="89" left="48" top="300.75" width="213" height="13.5" align-x="48" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Выберем произвольно первый вектор:</f> </text> </region> <region region-id="90" left="288" top="300.75" width="42" height="15.75" align-x="298.5" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="91" left="354" top="300.75" width="54" height="17.25" align-x="381" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIConstant" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="92" left="438" top="300.75" width="51" height="17.25" align-x="467.25" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>1</ml:real> </result> </ml:eval> </math> <rendering item-idref="9"/> </region> <region region-id="94" left="48" top="330" width="371.25" height="17.25" align-x="48" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Остальные векторы<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>определяются по следующему алгоритму.</f> </text> </region> <region region-id="41" left="60" top="360.75" width="287.25" height="165.75" align-x="75" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">E</ml:id> <ml:program> <ml:for> <ml:id xml:space="preserve">k</ml:id> <ml:range> <ml:real>2</ml:real> <ml:id xml:space="preserve">q</ml:id> </ml:range> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:localDefine> </ml:for> </ml:for> <ml:id xml:space="preserve">e</ml:id> </ml:program> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="75" left="78" top="555.75" width="104.25" height="57.75" align-x="93" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">E</ml:id> <ml:symResult> <ml:matrix rows="3" cols="4"> <ml:real>2</ml:real> <ml:real>3</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>2</ml:real> <ml:real>0</ml:real> <ml:real>3</ml:real> <ml:real>-3</ml:real> <ml:real>0</ml:real> <ml:real>-1</ml:real> <ml:real>-1</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="76" left="288" top="557.25" width="90" height="78.75" align-x="333" align-y="600" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:matcol/> <ml:parens> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:parens> <ml:real>1</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>2</ml:real> <ml:real>1</ml:real> <ml:real>3</ml:real> <ml:real>-1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="55" left="66" top="708.75" width="265.5" height="13.5" align-x="66" align-y="720" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <f family="Arial Cyr" charset="204" size="12">Проверим ортогональность полученного базиса:</f> </text> </region> <region region-id="64" left="66" top="744.75" width="33" height="15.75" align-x="82.5" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">w</ml:id> <ml:real>3</ml:real> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="63" left="114" top="744.75" width="30" height="15.75" align-x="127.5" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">v</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="65" left="186" top="729.75" width="117" height="50.25" align-x="281.25" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">j</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">w</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">E</ml:id> <ml:sequence> <ml:id xml:space="preserve">v</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0</ml:real> </result> </ml:eval> </math> <rendering item-idref="15"/> </region> <region region-id="97" left="42" top="798" width="452.25" height="84" align-x="42" align-y="810" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Что и требовалось показать. Легко решает задачи такого класса при любом </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="12">Однако задачу<sp count="2"/>4.70 из [24] </f> <f family="Arial Cyr" charset="204" size="12">с нулем </f> <f family="Arial Cyr" charset="204" size="12">не решает. Для этого нужен универсальный</f> <f family="Arial Cyr" charset="204" size="12">алгоритм приводимый ниже. Или всякий раз, когда </f> <f family="Arial Cyr" charset="204" size="12">появляется нулевая строка в матрице </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> с номером </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">, необходимо перемещать </inlineAttr> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">в матрице </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12">вектор с номером </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> q </f> <inlineAttr font-style="normal"> <f family="Arial Cyr" charset="204" size="12"> в последнюю строку и повторять вычисления. Вычисления прекратить, вместе </f> </inlineAttr> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">с перебором всех нижних </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> q </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">,...,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> <sp count="4"/>строк матрицы </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">f</f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="120" left="18" top="906.75" width="483" height="27.75" align-x="18" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">Рассмотрим процесс ортогонализации Шмидта,</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">учитывающий возможную линейную зависимость векторов. </inlineAttr> </f> </text> </region> <region region-id="121" left="18" top="957" width="445.5" height="34.5" align-x="18" align-y="972" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">В пространстве </f> <f family="Arial Cyr" charset="204" size="12"> R </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12"> задана произвольная система векторов </f> <f family="Arial Cyr" charset="204" size="12">(</f> <f family="Arial Cyr" charset="204" size="12">f</f> <f family="Arial Cyr" charset="204" size="12"> 1 </f> <f family="Arial Cyr" charset="204" size="12">,</f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> 2 </f> <f family="Arial Cyr" charset="204" size="12">,..., </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> n </f> <f family="Arial Cyr" charset="204" size="12">)</f> <f family="Arial Cyr" charset="204" size="12">, которые </f> <f family="Arial Cyr" charset="204" size="12">ниже записаны в строки матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12"> (см. </f> <f family="Arial Cyr" charset="204" size="12">задачу 4.70</f> <f family="Arial Cyr" charset="204" size="12"> из [24]).</f> </text> </region> <region region-id="122" left="24" top="1044.75" width="30" height="15.75" align-x="37.5" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" 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Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">df02640a8fdde5fdfd6e7b117dbabd17</contentHash> <ml:apply> <ml:matcol/> <ml:id xml:space="preserve">f</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>2</ml:real> <ml:real>7</ml:real> <ml:real>1</ml:real> <ml:real>5</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="19"> <element-image-map> <box left="1.5" top="27.75" width="15.75" height="18.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="126" left="18" top="1110.75" width="473.25" height="45" align-x="18" align-y="1122" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Запишем более общий алгоритм построения в этом же пространстве ортогонального базиса векторов {</f> <f family="Arial Cyr" charset="204" size="12"> е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}, который будет учитывать возможную линейную зависимость строк матрицы </f> <f family="Arial Cyr" charset="204" size="12"> f </f> <f family="Arial Cyr" charset="204" size="12">.</f> </text> </region> <region region-id="155" left="18" top="1170.75" width="243.75" height="13.5" align-x="18" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Итак, выберем произвольно первый вектор:</f> </text> </region> <region region-id="156" left="288" top="1170.75" width="42" height="15.75" align-x="298.5" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" 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align-x="491.25" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">7680f4a2f8d24ecee199dd71e66d8dfc</contentHash> <ml:define> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:real>1</ml:real> </ml:define> </ml:provenance> </math> <rendering item-idref="23"> <element-image-map> <box left="1.5" top="0.75" width="32.25" height="15.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="160" left="24" top="1206" width="467.25" height="17.25" align-x="24" align-y="1218" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Последующие векторы<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>определяются по алгоритму, в котором есть оператор </f> <f family="Arial Cyr" charset="204" size="12"> if </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="133" left="24" top="1236.75" width="439.5" height="362.25" align-x="39" align-y="1248" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" 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<ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:localDefine> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:localDefine> <ml:ifThen> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> </ml:ifThen> </ml:program> </ml:for> <ml:ifThen> <ml:apply> <ml:and/> <ml:apply> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:lessThan/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> </ml:apply> <ml:for> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:range> <ml:program> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">f</ml:id> <ml:sequence> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">m</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">m</ml:id> </ml:sequence> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:id xml:space="preserve">n</ml:id> </ml:bounds> </ml:apply> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">s</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> </ml:lambda> <ml:bounds> <ml:real>1</ml:real> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:apply> </ml:bounds> </ml:apply> </ml:apply> </ml:localDefine> <ml:ifThen> <ml:apply> <ml:greaterThan/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:sequence> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:localDefine> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">r</ml:id> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:localDefine> </ml:ifThen> </ml:program> </ml:for> </ml:ifThen> </ml:program> </ml:for> <ml:id xml:space="preserve">e</ml:id> </ml:program> </ml:define> </ml:provenance> </math> <rendering item-idref="24"> <element-image-map> <box left="1.5" top="0.75" width="438" height="360.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="134" left="36" top="1632" width="331.5" height="14.25" align-x="36" align-y="1644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Полученный ортогональный базис представлен матрицей </f> <f family="Arial Cyr" charset="204" size="12"> Е </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">. </inlineAttr> </f> </text> </region> <region region-id="135" left="60" top="1667.25" width="104.25" height="78.75" align-x="75" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">d99a8458601faf3c014d6cc7c3941186</contentHash> <ml:id xml:space="preserve">E</ml:id> </ml:provenance> <ml:symResult> <ml:matrix rows="4" cols="4"> <ml:real>2</ml:real> <ml:real>3</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>2</ml:real> <ml:real>5</ml:real> <ml:real>0</ml:real> <ml:real>3</ml:real> <ml:real>-3</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>-1</ml:real> <ml:real>-1</ml:real> <ml:real>10</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="25"> <element-image-map> <box left="1.5" top="32.25" width="6" height="14.25" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="136" left="210" top="1667.25" width="90" height="78.75" align-x="255" align-y="1710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:provenance expr-id="1"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </originRef> <parentRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and Settings\Konstantin\Мои документы\Компьют технол\ЛинАлгебра\ОртогонШм1.xmcd" xmlns="http://schemas.mathsoft.com/provenance10"> <hash/> </parentRef> <comment xmlns="http://schemas.mathsoft.com/provenance10"/> <originComment xmlns="http://schemas.mathsoft.com/provenance10"/> <contentHash xmlns="http://schemas.mathsoft.com/provenance10">a084419d3498ee28d7c049634be7b81f</contentHash> <ml:apply> <ml:matcol/> <ml:parens> <ml:apply> <ml:transpose/> <ml:id xml:space="preserve">E</ml:id> </ml:apply> </ml:parens> <ml:real>1</ml:real> </ml:apply> </ml:provenance> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="4" cols="1"> <ml:real>2</ml:real> <ml:real>1</ml:real> <ml:real>3</ml:real> <ml:real>-1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="26"> <element-image-map> <box left="1.5" top="21.75" width="36.75" height="24.75" expr-idref="1" xmlns="http://schemas.mathsoft.com/worksheet20"/> </element-image-map> </rendering> </region> <region region-id="137" left="48" top="1770.75" width="265.5" height="13.5" align-x="48" align-y="1782" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Arial Cyr" charset="204" size="12">Проверим ортогональность полученного базиса:</f> </text> </region> <region region-id="138" left="48" top="1806.75" width="33" height="15.75" align-x="64.5" align-y="1818" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:provenance expr-id="1" xmlns:ml="http://schemas.mathsoft.com/math20"> <originRef doc-id="56eb9e5a-b101-495c-906c-636909fdbf2b" version-id="72819622-10d8-487c-b7c6-74201e970b78" branch-id="00000000-0000-0000-0000-000000000000" revision-num="6" is-modified="false" region-id="0" href="C:\Documents and 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