Matrix/K^n to K^m/Linear mapping/Example

The mapping

${\displaystyle K^{n}\longrightarrow K^{m},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto M{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}s_{j}\\\sum _{j=1}^{n}a_{2j}s_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}s_{j}\end{pmatrix}},}$

which is given by an ${\displaystyle {}m\times n}$-matrix ${\displaystyle {}M=(a_{ij})_{1\leq i\leq m,\,1\leq j\leq n}}$, is linear.