Wedge product/Transformation with determinant/Fact

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{n}$ and ${}w_{1},\ldots ,w_{n}$ be vectors in ${}V$, fulfilling the relation

${}{\begin{pmatrix}v_{1}\\\vdots \\v_{n}\end{pmatrix}}=M{\begin{pmatrix}w_{1}\\\vdots \\w_{n}\end{pmatrix}}\,,$

where ${}M$ is an ${}n\times n$-matrix.

Then we have in ${}\bigwedge ^{n}V$ the identity

${}v_{1}\wedge \ldots \wedge v_{n}=(\det M)w_{1}\wedge \ldots \wedge w_{n}\,.$

Proof, Write another proof