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Cyr" charset="204" size="10">Д.з., часть 1.</f> </text> </region> <region region-id="48" left="30" top="54" width="451.5" height="30.75" align-x="30" align-y="66" 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">Пусть заданы базисы {</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}, {</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} и векторы </f> <f family="Arial Cyr" charset="204" size="12">х </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12">, имеющие следующие разложения по </f> <f family="Arial Cyr" charset="204" size="12">этим базисам: </f> </text> </region> <region region-id="11" left="54" top="99.75" width="78.75" height="50.25" align-x="66.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="2"/> </region> <region region-id="12" left="180" top="99.75" width="80.25" height="50.25" align-x="192.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="3"/> </region> <region region-id="19" left="48" top="174" width="326.25" height="17.25" align-x="48" align-y="186" 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">Зададим разложение базисных векторов </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>по базису<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region 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<ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="45" left="54" top="450" width="453.75" height="65.25" align-x="54" align-y="462" 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">Предположим вектор </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">задан в одном из базисов, например, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}, а вектор </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> задан в другом базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">i</f> <f family="Arial Cyr" charset="204" size="12">}.</f> <f family="Arial Cyr" charset="204" size="12"> Требуется найти их координаты в противоположных базисах.</f> <f family="Arial Cyr" charset="204" size="12">В</f> <f family="Arial Cyr" charset="204" size="12">ектор </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">в базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12">k</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} обозначим как</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12">, а </f> <f family="Arial Cyr" charset="204" size="12">вектор </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12">e</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} - как </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">. Тогда<sp count="2"/>для вычисления </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp count="2"/>и<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> в</f> <f family="Arial Cyr" charset="204" size="12">оспользуемся формулами</f> </text> </region> <region region-id="64" left="222" top="540.75" width="113.25" height="40.5" align-x="222" align-y="552" 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">Проверка обратного</f> <f family="Arial Cyr" charset="204" size="12">отображения</f> <f family="Arial Cyr" charset="204" size="12"> T </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="62" left="66" top="548.25" width="58.5" height="20.25" align-x="85.5" align-y="564" 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">xk</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">T</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="61" left="144" top="536.25" width="62.25" height="48.75" align-x="162.75" align-y="564" 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:id xml:space="preserve">xk</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>0</ml:real> <ml:real>2</ml:real> <ml:real>-1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="60" left="378" top="536.25" width="72.75" height="48.75" align-x="407.25" align-y="564" 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:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">xk</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>-1</ml:real> <ml:real>2</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="86" left="228" top="624.75" width="113.25" height="43.5" align-x="228" align-y="636" 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">Проверка обратного</f> <f family="Arial Cyr" charset="204" size="12">отображения</f> <f family="Arial Cyr" charset="204" size="12"> T </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"> ye </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="80" left="60" top="636.75" width="45.75" height="15.75" align-x="78.75" align-y="648" 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">ye</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="81" left="126" top="620.25" width="70.5" height="48.75" align-x="144" align-y="648" 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:id xml:space="preserve">ye</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>0.41421356237309515</ml:real> <ml:real>1</ml:real> <ml:real>3.4142135623730949</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="83" left="384" top="613.5" width="114.75" height="62.25" align-x="446.25" align-y="648" 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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">T</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>-0.99999999999999989</ml:real> <ml:real>2</ml:real> <ml:real>1.4142135623730949</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="87" left="54" top="708.75" width="342.75" height="13.5" align-x="54" 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">Доказательство последней формулы может быть следующим. </f> </text> </region> <region region-id="110" left="60" top="756.75" width="66" height="13.5" align-x="60" align-y="768" 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">1. Возьмем:</f> </text> </region> <region region-id="109" left="156" top="741.75" width="80.25" height="50.25" align-x="168.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="18"/> </region> <region region-id="108" left="252" top="756.75" width="66" height="13.5" align-x="252" align-y="768" 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">2. Возьмем:</f> </text> </region> <region region-id="107" left="348" top="741.75" width="89.25" height="50.25" align-x="364.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</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:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="19"/> </region> <region region-id="113" left="60" top="810" width="310.5" height="17.25" align-x="60" align-y="822" 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">3. Подставим в (1) вместо </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12"> его выражение (2). Получим</f> </text> </region> <region region-id="116" left="78" top="843.75" width="284.25" height="50.25" align-x="227.25" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">ye</ml:id> <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">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</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:id xml:space="preserve">i</ml:id> </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: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:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:parens> <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">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </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:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="20"/> </region> <region region-id="127" left="72" top="936.75" width="63.75" height="13.5" align-x="72" align-y="948" 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">Выражение</f> </text> </region> <region region-id="128" left="156" top="921.75" width="110.25" height="50.25" align-x="229.5" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <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">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </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:indexer/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="130" left="72" top="984" width="435" height="31.5" align-x="72" align-y="996" 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">есть матрица-столбец </f> <f family="Arial Cyr" charset="204" size="12"> T </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">, компоненты которой являются координатами век-</f> <f family="Arial Cyr" charset="204" size="12">тора </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в базисе </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}. 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<center>{\rtf1\ansi\ansicpg1251\deff0\deflang1049{\fonttbl{\f0\fnil\fprq15\fcharset204{\*\fname Arial;}Arial CYR;}} \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="0" 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="130" 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="24" top="12.75" width="314.25" height="26.25" align-x="24" align-y="24" 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"> Определение координат вектора в разных базисах </f> <f family="Arial Cyr" charset="204" size="12">. </f> <f family="Arial Cyr" charset="204" size="10">Д.з., часть 1.</f> </text> </region> <region region-id="48" left="30" top="54" width="451.5" height="30.75" align-x="30" align-y="66" 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">Пусть заданы базисы {</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}, {</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} и векторы </f> <f family="Arial Cyr" charset="204" size="12">х </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">и</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12">, имеющие следующие разложения по </f> <f family="Arial Cyr" charset="204" size="12">этим базисам: </f> </text> </region> <region region-id="11" left="54" top="99.75" width="78.75" height="50.25" align-x="66.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">x</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">a</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="2"/> </region> <region region-id="12" left="180" top="99.75" width="80.25" height="50.25" align-x="192.75" align-y="126" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="3"/> </region> <region region-id="19" left="48" top="174" width="326.25" height="17.25" align-x="48" align-y="186" 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">Зададим разложение базисных векторов </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp count="2"/>по базису<sp count="2"/> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> </text> </region> <region region-id="25" left="402" top="159.75" width="89.25" height="50.25" align-x="418.5" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</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:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="4"/> </region> <region region-id="27" left="48" top="222" width="449.25" height="37.5" align-x="48" align-y="234" 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">Таким образом, запишем матрицу перехода </f> <f family="Arial Cyr" charset="204" size="12"> T </f> <f family="Arial Cyr" charset="204" size="12"> от базиса </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} к базису</f> <f family="Arial Cyr" charset="204" size="12"> {</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} размером</f> <f family="Arial Cyr" charset="204" size="14"> n </f> <f family="Arial Cyr" charset="204" size="12">x</f> <f family="Arial Cyr" charset="204" size="14">n</f> <f family="Arial Cyr" charset="204" size="14"> <inlineAttr font-style="normal">, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">в которой </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> j </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">-й столбец имеет компоненты t</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal"> i,j </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">.</inlineAttr> </f> </text> </region> <region region-id="30" left="102" top="278.25" width="72" height="48.75" align-x="117" align-y="306" 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="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">T</ml:id> <ml:matrix rows="3" cols="3"> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="49" left="258" top="294.75" width="45" height="15.75" align-x="282.75" align-y="306" 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:determinant/> <ml:id xml:space="preserve">T</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>2</ml:real> </result> </ml:eval> </math> <rendering item-idref="6"/> </region> <region region-id="35" left="48" top="348.75" width="352.5" height="13.5" align-x="48" align-y="360" 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">Зададим необходимые для дальнейших вычислений параметры:</f> </text> </region> <region region-id="36" left="54" top="396.75" width="28.5" height="15.75" align-x="67.5" align-y="408" 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>3</ml:real> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="37" left="108" top="380.25" width="51.75" height="48.75" align-x="121.5" align-y="408" 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">a</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>-1</ml:real> <ml:real>2</ml:real> <ml:real>1</ml:real> </ml:matrix> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="38" left="192" top="379.5" width="54.75" height="51" align-x="206.25" align-y="408" 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">b</ml:id> <ml:matrix rows="3" cols="1"> <ml:real>-1</ml:real> <ml:real>2</ml:real> <ml:apply> <ml:sqrt/> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="75" left="306" top="402.75" width="44.25" height="17.25" align-x="323.25" align-y="414" 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">a</ml:id> <ml:real>0</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>-1</ml:real> </result> </ml:eval> </math> <rendering item-idref="10"/> </region> <region region-id="76" left="366" top="385.5" width="41.25" height="34.5" align-x="384.75" align-y="414" 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:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:pow/> <ml:real>2</ml:real> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="45" left="54" top="450" width="453.75" height="65.25" align-x="54" align-y="462" 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">Предположим вектор </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-style="normal">задан в одном из базисов, например, </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}, а вектор </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> задан в другом базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12">i</f> <f family="Arial Cyr" charset="204" size="12">}.</f> <f family="Arial Cyr" charset="204" size="12"> Требуется найти их координаты в противоположных базисах.</f> <f family="Arial Cyr" charset="204" size="12">В</f> <f family="Arial Cyr" charset="204" size="12">ектор </f> <f family="Arial Cyr" charset="204" size="12"> х </f> <f family="Arial Cyr" charset="204" size="12">в базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12">k</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} обозначим как</f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12">, а </f> <f family="Arial Cyr" charset="204" size="12">вектор </f> <f family="Arial Cyr" charset="204" size="12"> у </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в</inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> <sp/> </f> <f family="Arial Cyr" charset="204" size="12">базисе </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12">e</f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">} - как </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal">. Тогда<sp count="2"/>для вычисления </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp count="2"/>и<sp count="2"/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal"> <sp/> </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12"> в</f> <f family="Arial Cyr" charset="204" size="12">оспользуемся формулами</f> </text> </region> <region region-id="64" left="222" top="540.75" width="113.25" height="40.5" align-x="222" align-y="552" 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">Проверка обратного</f> <f family="Arial Cyr" charset="204" size="12">отображения</f> <f family="Arial Cyr" charset="204" size="12"> T </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> xk </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> a </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="62" left="66" top="548.25" width="58.5" height="20.25" align-x="85.5" align-y="564" 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">xk</ml:id> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">T</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:id xml:space="preserve">a</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="61" left="144" top="536.25" width="62.25" height="48.75" align-x="162.75" align-y="564" 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:id xml:space="preserve">xk</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>0</ml:real> <ml:real>2</ml:real> <ml:real>-1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="13"/> </region> <region region-id="60" left="378" top="536.25" width="72.75" height="48.75" align-x="407.25" align-y="564" 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:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">xk</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>-1</ml:real> <ml:real>2</ml:real> <ml:real>1</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="14"/> </region> <region region-id="86" left="228" top="624.75" width="113.25" height="43.5" align-x="228" align-y="636" 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">Проверка обратного</f> <f family="Arial Cyr" charset="204" size="12">отображения</f> <f family="Arial Cyr" charset="204" size="12"> T </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"> ye </f> <f family="Arial Cyr" charset="204" size="12">=</f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">:</f> </text> </region> <region region-id="80" left="60" top="636.75" width="45.75" height="15.75" align-x="78.75" align-y="648" 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">ye</ml:id> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="15"/> </region> <region region-id="81" left="126" top="620.25" width="70.5" height="48.75" align-x="144" align-y="648" 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:id xml:space="preserve">ye</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>0.41421356237309515</ml:real> <ml:real>1</ml:real> <ml:real>3.4142135623730949</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="16"/> </region> <region region-id="83" left="384" top="613.5" width="114.75" height="62.25" align-x="446.25" align-y="648" 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:mult style="default"/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">T</ml:id> <ml:real>-1</ml:real> </ml:apply> <ml:matrix rows="3" cols="1"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>1</ml:real> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ye</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:matrix> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:matrix rows="3" cols="1"> <ml:real>-0.99999999999999989</ml:real> <ml:real>2</ml:real> <ml:real>1.4142135623730949</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="17"/> </region> <region region-id="87" left="54" top="708.75" width="342.75" height="13.5" align-x="54" 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">Доказательство последней формулы может быть следующим. </f> </text> </region> <region region-id="110" left="60" top="756.75" width="66" height="13.5" align-x="60" align-y="768" 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">1. Возьмем:</f> </text> </region> <region region-id="109" left="156" top="741.75" width="80.25" height="50.25" align-x="168.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:id xml:space="preserve">y</ml:id> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="18"/> </region> <region region-id="108" left="252" top="756.75" width="66" height="13.5" align-x="252" align-y="768" 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">2. Возьмем:</f> </text> </region> <region region-id="107" left="348" top="741.75" width="89.25" height="50.25" align-x="364.5" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">k</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</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:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="19"/> </region> <region region-id="113" left="60" top="810" width="310.5" height="17.25" align-x="60" align-y="822" 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">3. Подставим в (1) вместо </f> <f family="Arial Cyr" charset="204" size="12"> k </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12"> его выражение (2). Получим</f> </text> </region> <region region-id="116" left="78" top="843.75" width="284.25" height="50.25" align-x="227.25" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:equal/> <ml:id xml:space="preserve">ye</ml:id> <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">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</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:id xml:space="preserve">i</ml:id> </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: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:apply> <ml:summation/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">i</ml:id> </ml:boundVars> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:parens> <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">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </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:parens> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">e</ml:id> <ml:id xml:space="preserve">i</ml:id> </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> </math> <rendering item-idref="20"/> </region> <region region-id="127" left="72" top="936.75" width="63.75" height="13.5" align-x="72" align-y="948" 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">Выражение</f> </text> </region> <region region-id="128" left="156" top="921.75" width="110.25" height="50.25" align-x="229.5" align-y="948" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <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">t</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">b</ml:id> <ml:id xml:space="preserve">j</ml:id> </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:indexer/> <ml:parens> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">T</ml:id> <ml:id xml:space="preserve">b</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">i</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="130" left="72" top="984" width="435" height="31.5" align-x="72" align-y="996" 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">есть матрица-столбец </f> <f family="Arial Cyr" charset="204" size="12"> T </f> <f family="Arial Cyr" charset="204" size="12">*</f> <f family="Arial Cyr" charset="204" size="12"> b </f> <f family="Arial Cyr" charset="204" size="12">, компоненты которой являются координатами век-</f> <f family="Arial Cyr" charset="204" size="12">тора </f> <f family="Arial Cyr" charset="204" size="12"> уе </f> <f family="Arial Cyr" charset="204" size="12"> <inlineAttr font-weight="normal" font-style="normal">в базисе </inlineAttr> </f> <f family="Arial Cyr" charset="204" size="12">{</f> <f family="Arial Cyr" charset="204" size="12"> e </f> <f family="Arial Cyr" charset="204" size="12"> i </f> <f family="Arial Cyr" charset="204" size="12">}. 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