TITLE: Every Matrix a linear transformation?
QUESTION [5 upvotes]: Every finite-dimensional linear map can be represented by a matrix. But what about the opposite: Does every matrix correspond to a linear map?

REPLY [12 votes]: 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.