Surjective and full rank

Question

We have a linear mapping $f: \mathbb{R^n}\to\mathbb{R^m}$, $x \mapsto Cx$, defined by a matrix $C\in\mathbb{R}^{m\times n}$ with $m \leq n$. How can I show that $f$ being surjective is equivalent to the matrix $C$ having full rank.

Answer 1

Hint: $f$ is surjective if and only if for every vector $y \in \Bbb R^m$, the system of equations $Cx = y$ has at least one solution. Based on what you know about solving systems of equations: if every choice of $y$ yields a solution, what can you deduce about the matrix $C$?