Do all n x n matrices over the reals represent linear transformations?

Question

Do all $v \in M_n (\mathbb{R})$ represent linear transformations?

To add to that a bit to further clarify for myself:

Looking up the def. of a transformation it is any function $f$ mapping a set $X$ to itself, i.e., $f:X →X$. If it is linear then $f(av+bw) = af(v) + bf(w)$. So is $f: \mathbb{R}^3 → \mathbb{R}$ not a transformation? If not, what is it?

Answer 1

Given any matrix $M\in \mathbb R^{m\times n}$ define a function $T:\mathbb R^n\to \mathbb R^m$ as $T(v)=Mv$ for all $v\in \mathbb R^n$. It is your exercise to show that $T$ is well-defined (hence a function) and a linear transformation.