Every finite-dimensional linear map can be represented by a matrix. But what about the opposite: Does every matrix correspond to a linear map?
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.