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.

${}\varphi$ is injective if and only if the columns of the matrix are linearly independent.
${}\varphi$ is surjective if and only if the columns of the matrix form a generating system of ${}K^{m}$ .
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.