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Determinant via multilinear functionals
Definition
$V$ … finite dimensional $\mathcal F$-vector space |
$\mathrm{det}:L(V,V)\to \mathcal F$ |
$n\equiv \mathrm{dim}(V)$ |
$M\in \mathrm{MultiLin}(V^n)$ |
$ v_1,\dots,v_n\in V $ |
$A\in L(V,V)$ |
$ M(A\ v_1,\dots,A\ v_n) = \mathrm{det}(A)\cdot M(v_1,\dots,v_n) $ |
Discussion
Theorems
- The determinant is an invariant of linear operators on finite-dimensional vector spaces.
- $\mathrm{det}(AB)=\mathrm{det}(A)\cdot \mathrm{det}(B)$
- $\mathrm{det}(Id)=1$
- $\mathrm{det}(A)\ne 0$ is $A$ is a linear isomorphism
- $\mathrm{det}(A)\ne 0\Rightarrow \mathrm{det}(A^{-1})=\mathrm{det}(A)^{-1}$
Reference
Wikipedia: Determinant