===== Determinant via multilinear functionals =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $V$ ... finite dimensional $\mathcal F$-vector space | 

| @#FFBB00: definiendum | @#FFBB00: $\mathrm{det}:L(V,V)\to \mathcal F$ |

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

| @#55EE55: postulate   | @#55EE55: $ 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: [[http://en.wikipedia.org/wiki/Determinant|Determinant]]
==== Parents ====
=== Context ===
[[Multilinear functional]], [[Linear operator algebra]], [[Vector space dimension]]