Every Matrix a linear transformation?
Every finite-dimensional linear map can be represented by a matrix. But what about the opposite: Does every matrix correspond to a linear map?
$\endgroup$ 33 Answers
$\begingroup$Well, not every matrix, necessarily. We could take matrices of arbitrary sets, for example, which needn't have any associated operations.
However, assuming that you're taking matrices of elements of some field $\Bbb F,$ then the answer is yes. Given any such matrix, say $A,$ if $A$ is $m\times n,$ then the map $T:\Bbb F^n\to\Bbb F^m$ given by $T(\vec x)=A\vec x$ is linear.
$\endgroup$ 5 $\begingroup$Yes. If you have a $m\times n$ matrix $M$, then this can be seen as a map from $\mathbb{R}^n$ to $\mathbb{R}^m$ by $M(x) = Mx$.
$\endgroup$ $\begingroup$If $A\in\mathcal M_{n,p}(\Bbb R)$ then the map $$f\colon \Bbb R^p\rightarrow \Bbb R^n,\quad x\mapsto A x$$ is a linear transformation which's represented by the matrix $A$ in the canonical basis.
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