Vector space/mxn-Matrices/Example

For a field ${}K$, and given natural numbers ${}m,n$, the set

$\operatorname {Mat} _{m\times n}(K)$

of all ${}m\times n$-matrices with componentwise addition and componentwise scalar multiplication, is a ${}K$-vector space. The null element in this vector space is the null matrix

${}0={\begin{pmatrix}0&\ldots &0\\\vdots &\ddots &\vdots \\0&\ldots &0\end{pmatrix}}\,.$