Linear mapping/Determinant/Introduction/Section

Let

$\varphi \colon V\longrightarrow V$

be a linear mapping from a vector space of dimension ${}n$ into itself. This is described by a matrix ${}M\in \operatorname {Mat} _{n}(K)$ 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 ${}M$ and ${}N$, and the matrix ${}B$ for the change of bases, we have the relation ${}N=BMB^{-1}$. The multiplication theorem for determinants yields then

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

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

Definition

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

$\varphi \colon V\longrightarrow V$

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

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

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