<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 </xsl:when>
<xsl:otherwise>is not </xsl:otherwise>
</xsl:choose>
a basis of <m>\mathbb{R}^<xsl:value-of select="dim"/></m>.
</claim>
</p>
<ol>
<li> Write an equivalent statement in terms of other vector properties.</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 </xsl:when>
<xsl:otherwise>is not </xsl:otherwise>
</xsl:choose>
a basis of <m>\mathbb{R}^<xsl:value-of select="dim"/></m>.
</claim>
is equivalent to the statement
<claim> The set of vectors <m><xsl:value-of select="vecset"/></m>
<xsl:choose>
<xsl:when test="prompt">both spans <m>\mathbb{R}^<xsl:value-of select="dim"/></m> and is linearly independent.</xsl:when>
<xsl:otherwise>either doesn't span <m>\mathbb{R}^<xsl:value-of select="dim"/></m> or is linearly dependent.</xsl:otherwise>
</xsl:choose>
</claim>
</li>
<li>The set of vectors <m><xsl:value-of select="vecset"/></m>
<xsl:choose>
<xsl:when test="basis"> is </xsl:when>
<xsl:otherwise> is not </xsl:otherwise>
</xsl:choose>
a basis of <m>\mathbb{R}^<xsl:value-of select="dim"/></m>
</li>
</ol>
</answer>
</exercise>
