Linear mapping/Matrix to basis/Several properties/Fact

Let $K$ be a field, and let $V$ and $W$ be vector spaces over $K$ of dimensions $n$ and $m$. Let

$\phi :V⟶W$

be a linear map, described by the matrix $M\in {\mathrm{Mat}}_{m\times n}(K)$ with respect to two bases. Then the following properties hold.

1. $\phi$ is injective if and only if the columns of the matrix are linearly independent.
2. $\phi$ is surjective if and only if the columns of the matrix form a generating system of $K^m$.
3. Let $m=n$. Then $\phi$ is bijective if and only if the columns of the matrix form a basis of $K^m$, and this holds if and only if $M$ is invertible.

Proof, Write another proof