<exercise>
<statement>
<p>Consider the statement
<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>
<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>
</claim>
</p>
<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>The statement
<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>
<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>
</claim>
is equivalent to the statement
<claim>
<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>
</claim>
</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>
