<exercise>
<statement>
<p>Consider the statement
<claim> The set of vectors <m><xsl:value-of select="vecset"/></m>
<xsl:choose>
<xsl:when test="prompt">is linearly independent.</xsl:when>
<xsl:otherwise>is linearly dependent.</xsl:otherwise>
</xsl:choose>
</claim>
</p>
<ol>
<li> Write an equivalent statement using a vector 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 vectors <m><xsl:value-of select="vecset"/></m>
<xsl:choose>
<xsl:when test="prompt">is linearly independent.</xsl:when>
<xsl:otherwise>is linearly dependent.</xsl:otherwise>
</xsl:choose>
</claim>
is equivalent to the statement
<claim> The vector equation <m><xsl:value-of select="veceq"/></m>
<xsl:choose>
<xsl:when test="prompt">has no nontrivial solutions.</xsl:when>
<xsl:otherwise>has (infinitely many) nontrivial solutions.</xsl:otherwise>
</xsl:choose>
</claim>
</li>
<li> The set of vectors <m><xsl:value-of select="vecset"/></m>
<xsl:choose>
<xsl:when test="independent">is linearly independent.</xsl:when>
<xsl:otherwise>is linearly dependent.</xsl:otherwise>
</xsl:choose>
</li>
</ol>
</answer>
</exercise>
