Endomorphism/Determinant/Definition

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ denote a ${\displaystyle {}K}$-vector space of finite dimension. Let

${\displaystyle \varphi \colon V\longrightarrow V}$

be a linear mapping, which is described by the matrix ${\displaystyle {}M}$, with respect to a basis. Then

${\displaystyle {}\det \varphi$ :=\det M\,}

is called the determinant of the linear mapping ${\displaystyle {}\varphi }$.