Linear mapping/Determinant/Introduction/Section

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be a linear mapping from a vector space of dimension into itself. This is described by a matrix 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 and , and the matrix for the change of bases, we have the relation . The multiplication theorem for determinants yields then

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

Let denote a field, and let denote a -vector space of finite dimension. Let

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

is called the determinant of the linear mapping .