Vector space/Orientation/Wedge product/Fact/Proof

Proof

Let ${}v_{1},\ldots ,v_{n}$ and ${}w_{1},\ldots ,w_{n}$ be two bases of ${}V$, fulfilling the relation

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

Due to fact, we have

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

This shows the well-definedness of the mapping, and the statement follows.

To fact