<exercise>
<statement>
Let <m>T:\mathbb{R}^<xsl:value-of select="columns"/> \to \mathbb{R}^<xsl:value-of select="rows"/></m> be the linear transformation given by
<me>T\left( <xsl:value-of select="varvector"/> \right) = <xsl:value-of select="Tvar"/>.</me>
<ol>
<li>Explain how to find the image of <m>T</m> and the kernel of <m>T</m>.</li>
<li>Explain how to find a basis of the image of <m>T</m> and a basis of the kernel of <m>T</m>.</li>
<li>Explain how to find the rank and nullity of <m>T</m>, and why the rank-nullity theorem holds for <m>T</m>.</li>
</ol>
</statement>
<answer>
<p><me>\operatorname{RREF}<xsl:value-of select="matrix"/>=<xsl:value-of select="rref"/></me></p>
<ol>
<li>
<me>\operatorname{Im}\ T = \operatorname{span}\ <xsl:value-of select="imagebasis"/></me>
<me>\operatorname{ker}\ T = <xsl:value-of select="kernel"/></me>
</li>
<li>
A basis of <m>\operatorname{Im}\ T</m> is <m><xsl:value-of select="imagebasis"/></m>.
A basis of <m>\operatorname{ker}\ T</m> is <m><xsl:value-of select="kernelbasis"/></m>
</li>
<li>
The rank of <m>T</m> is <m><xsl:value-of select="rank"/></m>, the nullity of <m>T</m> is <m><xsl:value-of select="nullity"/></m>,
and the dimension of the domain of <m>T</m> is <m><xsl:value-of select="columns"/></m>. The rank-nullity theorem asserts that
<m><xsl:value-of select="rank"/>+<xsl:value-of select="nullity"/>=<xsl:value-of select="columns"/></m>, which we see to be true.
</li>
</ol>
</answer>
</exercise>
