Real vector space/Bijective/Orientation-preserving/Positive determinant/Fact

Let ${}V$ be a finite-dimensional real vector space, and let ${}\varphi \colon V\rightarrow V$ denote a bijective linear mapping.

Then ${}\varphi$ is

orientation-preserving if and only if the determinant

of ${}\varphi$ is positive.

Proof, Write another proof