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
