# Linear mapping/Determinant/Introduction/Section

Let

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

be a linear mapping from a vector space of dimension ${\displaystyle {}n}$ into itself. This is described by a matrix ${\displaystyle {}M\in \operatorname {Mat} _{n}(K)}$ with respect to a given basis. We would like to define the determinant of the linear mapping, by the determinant of the matrix. However, we have here the problem whether this is well-defined, since a linear mapping is described by quite different matrices, with respect to different bases. But, because of fact, when we have two describing matrices ${\displaystyle {}M}$ and ${\displaystyle {}N}$, and the matrix ${\displaystyle {}B}$ for the change of bases, we have the relation ${\displaystyle {}N=BMB^{-1}}$. The multiplication theorem for determinants yields then

${\displaystyle {}\det N=\det {\left(BMB^{-1}\right)}=(\det B)(\det M){\left(\det B^{-1}\right)}=(\det B){\left(\det B^{-1}\right)}(\det M)=\det M\,,}$

so that the following definition is in fact independent of the basis chosen.

## 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 ${\displaystyle {}\varphi }$.