Vector space/Addition in coordinate system/Independence of basis/Exercise

Let ${}V$ be a ${}K$-vector space, let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, and let

$\psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V,{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n},$

be the corresponding bijective mapping in the sense of remark. Show that this mapping transforms the componentwise addition in ${}K^{n}$ into the vector addition in ${}V$, that is,

${}\psi _{\mathfrak {v}}{\left({\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\right)}=\psi _{\mathfrak {v}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+\psi _{\mathfrak {v}}{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\,$

holds.

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