CoCalc -- V8.ptx

²⁰⁸⁶⁵ views
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<exercise>
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  <statement>
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  <p>Consider the statement
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        <claim> The set of <xsl:choose><xsl:when test="context='polynomial'">polynomials </xsl:when><xsl:otherwise>matrices </xsl:otherwise></xsl:choose> <m><xsl:value-of select="set"/></m> 
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            <xsl:choose>
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                <xsl:when test="task='independent'"> is linearly independent.</xsl:when>
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                <xsl:otherwise> spans <xsl:choose><xsl:when test="context='polynomial'"><m>\mathcal{P}_3</m></xsl:when><xsl:otherwise><m>\mathrm{M}_{2,2}</m></xsl:otherwise></xsl:choose>. </xsl:otherwise>
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            </xsl:choose>
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        </claim>
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	</p>
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    <ol>
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        <li> Write an equivalent statement using a <xsl:choose><xsl:when test="context='polynomial'">polynomial</xsl:when><xsl:otherwise>matrix</xsl:otherwise></xsl:choose> equation.</li>
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        <li> Explain why your statement is true or false.</li>
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    </ol>
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  </statement>
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  <answer>
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     <ol>
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          <li>The statement
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            <claim> The set of <xsl:choose><xsl:when test="context='polynomial'">polynomials </xsl:when><xsl:otherwise>matrices </xsl:otherwise></xsl:choose> <m><xsl:value-of select="set"/></m> 
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            <xsl:choose>
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                <xsl:when test="task='independent'"> is linearly independent.</xsl:when>
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                <xsl:otherwise> spans <xsl:choose><xsl:when test="context='polynomial'"><m>\mathcal{P}_3</m></xsl:when><xsl:otherwise><m>\mathrm{M}_{2,2}</m></xsl:otherwise></xsl:choose>. </xsl:otherwise>
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            </xsl:choose>
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        </claim>
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            is equivalent to the statement
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            <claim> 
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            <xsl:choose>
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                <xsl:when test="task='independent'">
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                    The <xsl:choose><xsl:when test="context='polynomial'">polynomial </xsl:when><xsl:otherwise>matrix </xsl:otherwise></xsl:choose> equation 
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                    <me><xsl:value-of select="equation"/></me>
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                    has no nontrivial solutions.
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                </xsl:when>
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                <xsl:otherwise> 
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                    <xsl:choose>
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                        <xsl:when test="context='polynomial'">
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                            The polynomial equation <me><xsl:value-of select="equation"/>=f</me> has a solution for every <m>f \in \mathcal{P}_3</m>.
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                        </xsl:when>
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                        <xsl:otherwise>
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                            The matrix equation <me><xsl:value-of select="equation"/>=B</me> has a solution for every <m>B \in \mathrm{M}_{2,2}</m>.
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                        </xsl:otherwise>
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                     </xsl:choose>
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                </xsl:otherwise>
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            </xsl:choose>
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            </claim>
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          </li>
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              <li>The set of <xsl:choose><xsl:when test="context='polynomial'">polynomials </xsl:when><xsl:otherwise>matrices </xsl:otherwise></xsl:choose> <m><xsl:value-of select="set"/></m> 
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            <xsl:choose>
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                <xsl:when test="task='independent'"> <xsl:choose><xsl:when test="result">is linearly independent</xsl:when><xsl:otherwise>is linearly dependent</xsl:otherwise></xsl:choose>.</xsl:when>
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                <xsl:otherwise> <xsl:choose><xsl:when test="result"> spans </xsl:when><xsl:otherwise> does not span </xsl:otherwise></xsl:choose>
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                <xsl:choose><xsl:when test="context='polynomial'"><m>\mathcal{P}_3</m></xsl:when><xsl:otherwise><m>\mathrm{M}_{2,2}</m></xsl:otherwise></xsl:choose>. </xsl:otherwise>
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            </xsl:choose>
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          </li>
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      </ol>
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  </answer>
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</exercise>
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