<?xml version='1.0' encoding='UTF-8'?>
<exercise xmlns="https://spatext.clontz.org" version="0.0">
<statement>
<p>
Let <m>T:\mathbb{R}^{{columns}} \to \mathbb{R}^{{rows}}</m>
be the linear transformation given by
<me>T\left( {{varvector}} \right) = {{Tvar}}.</me>
</p>
<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}{{matrix}}={{rref}}</me></p>
<ol>
<li>
<p>
<m>\operatorname{Im}\ T = {{image}}</m> and
<m>\operatorname{ker}\ T = {{kernel}}</m>
</p>
</li>
<li>
<p>
A basis of <m>\operatorname{Im}\ T</m> is <m>{{image_basis}}</m>.
A basis of <m>\operatorname{ker}\ T</m> is <m>{{kernel_basis}}</m>.
</p>
</li>
<li>
<p>
The rank of <m>T</m> is <m>{{rank}}</m>, the nullity of <m>T</m> is <m>{{nullity}}</m>,
and the dimension of the domain of <m>T</m> is <m>{{columns}}</m>. The rank-nullity theorem asserts that
<m>{{rank}}+{{nullity}}={{columns}}</m>, which we see to be true.
</p>
</li>
</ol>
</answer>
</exercise>
