Matrix/Linear mapping/Exercise

Let ${}M={\left(a_{ij}\right)}_{1\leq i\leq m,\,1\leq j\leq n}$ be an ${}m\times n$-matrix over a field ${}K$. Show that the corresponding mapping

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

is linear.

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