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AT4-Equation Mirror Observation #658

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AT4-Equation Mirror Observation #658

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20 changes: 20 additions & 0 deletions source/linear-algebra/source/03-AT/04.ptx
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Expand Up @@ -352,6 +352,26 @@ can never equal
</statement>
</activity>

<observation>
<statement>
<p>
Given a linear transformation <m>T\colon V\to W</m>, we can restate injectivity and surjectivity as statement about solutions to particular equations:
<ul>
<li>
<p>
The transformation <m>T</m> is injective precisely when the equation <m>T(\vec{x})=\vec{b}</m> has <em>at most</em> one solution for each <m>\vec{b}\in W</m>.
</p>
</li>
<li>
<p>
The transformation <m>T</m> is surjective precisely when the equation <m>T(\vec{x})=\vec{b}</m> has <em>at least</em> one solution for each <m>\vec{b}\in W</m>.
</p>
</li>
</ul>
</p>
</statement>
</observation>

<!-- <observation>
<p>
As we will see, it's no coincidence that the <m>\RREF</m> of the
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