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

$\varphi \colon V\longrightarrow W$

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

1. ${}\varphi$ is injective if and only if the columns of the matrix are linearly independent.
2. ${}\varphi$ is surjective if and only if the columns of the matrix form a generating system of ${}K^{m}$.
3. Let ${}m=n$. Then ${}\varphi$ 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