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$A$ is an m x n matrix:

$T(\vec{x})=A\vec{x}$

$ im(T) = {A\vec{x} | \vec{x} \epsilon \mathbb{R}^n } $ = column space of $A$

Can someone illustrate this fact with an example? (The fact that the $im(T)$ = column space of $A$

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1 Answer 1

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An example: consider the $3\times 2$ matrix $A = \left[\begin{array}{cc}1&0\\2&1\\0&1\end{array}\right]$. This defines the linear map $T:\mathbb{R}^2\to\mathbb{R}^3$ given by

$T\left[\begin{array}{c}x\\y\end{array}\right] = A\left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}x\\2x+y\\y\end{array}\right]$

We can rewrite this as

$T\left[\begin{array}{c}x\\y\end{array}\right] = x\left[\begin{array}{c}1\\2\\0\end{array}\right]+y\left[\begin{array}{c}0\\1\\1\end{array}\right]$, and so we see that the set of all vectors of the form $T(x,y)$ is exactly the span of the columns of $A$. That is, the image of $T$ is equal to the column space of $A$.

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