## 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