"*A transformation $T$ is linear if and only if it is a matrix transformation of the form $T=T_A$.*" Is this theorem true?

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Theorem:

Let $A$ be an $m \times n$ matrix, and $\mathbf x$ be a vector in $\mathbb R^n$. A transformation $T:\mathbb R^n \rightarrow \mathbb R^m$ is linear if and only if it is a matrix transformation of the form $T_A(\mathbf x)=A \mathbf x$, where $T=T_A$.

This theorem is not available in my linear algebra sources, but I think, by intuition, it should be true. Is it really true?

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1 Answer

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Yes. (Being a little bit pedantic, it is actually formulated incorrectly, but I know what you mean). I think you already know how to prove that a matrix transformation is linear, so that's one direction. For the other direction, you can construct the matrix: for $i=1,\dots,n$, the $i$th column of the matrix associated to $T$ is $T(e_i)$. Check that this results in the same transformation as $T$.

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