Let $(V_1, \F ), (V_2, \F )$ be two vector spaces and $f: V_1 \to V_2$ be a transformation between them.

If $f$ is linear, then there exists bases for the spaces and a matrix $A$ in $\F $ for which matrix multiplication is equivalent to applying $f$.

If $A$ is a matrix in $F$ $f$ is linear if and only if there exists a matrix in $\F $ whose coordinates in $V_1$ to coordinates in $V_2$.