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determinant_via_multilinear_functionals [2013/09/17 23:20] nikolaj old revision restored (2013/09/17 23:06) |
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| - | ===== Determinant via multilinear functional ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $V$ ... finite dimensional $\mathcal F$-vector space | | ||
| - | | @#FFBB00: $\mathrm{det}:L(V,V)\to \mathcal F$ | | ||
| - | |||
| - | | @#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: $ 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 ==== | ||
| - | === Requirements === | ||
| - | [[Multilinear functional]], [[Linear operator algebra]], [[Vector space dimension]] | ||
