# Basic Linear Algebra by Jürgen Müller By Jürgen Müller

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An ) ∆(a1 , . . , an ). 8) Determinants of linear maps. Let V be an R-vector space with finite R-basis B, and let ϕ : V → V be R-linear. Then det(ϕ) := det(B ϕB ) ∈ R is independent from the R-basis of V chosen, and is called the determinant of ϕ: Indeed, if C ⊆ V is an R-basis, then we have det(C ϕC ) = det(C idB · B ϕB · −1 · det(B ϕB ) · det(B idC ) = det(B ϕB ). B idC ) = det(B idC ) In particular, in view of the geometric interpretation of determinants, if V = Rn×1 and B ⊆ Rn×1 is the standard R-basis, then det(ϕ) is just the oriented volume of the image of the unit cube under ϕ, thus describes the change in volume application of the R-linear map ϕ entails.

N} (aj − a1 ) · ∆(a2 , . . , an ) ∆(a1 , . . , an ). 8) Determinants of linear maps. Let V be an R-vector space with finite R-basis B, and let ϕ : V → V be R-linear. Then det(ϕ) := det(B ϕB ) ∈ R is independent from the R-basis of V chosen, and is called the determinant of ϕ: Indeed, if C ⊆ V is an R-basis, then we have det(C ϕC ) = det(C idB · B ϕB · −1 · det(B ϕB ) · det(B idC ) = det(B ϕB ). B idC ) = det(B idC ) In particular, in view of the geometric interpretation of determinants, if V = Rn×1 and B ⊆ Rn×1 is the standard R-basis, then det(ϕ) is just the oriented volume of the image of the unit cube under ϕ, thus describes the change in volume application of the R-linear map ϕ entails.

Stroth: Lineare Algebra, 2. Auflage, Berliner Studienreihe zur Mathematik 7, Heldermann, 2008.