determinant_via_multilinear_functionals

Determinant via multilinear functionals

Set

context

$V$ … finite dimensional $\mathcal F$-vector space

definiendum

$\mathrm{det}:L(V,V)\to \mathcal F$

range	$n\equiv \mathrm{dim}(V)$
$M\in \mathrm{MultiLin}(V^n)$
$ v_1,\dots,v_n\in V $
$A\in L(V,V)$

postulate

$ 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

Parents

Context

Multilinear functional, Linear operator algebra, Vector space dimension