V8.ptx

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