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 determinant_via_multilinear_functionals [2013/09/17 23:20]nikolaj old revision restored (2013/09/17 23:06) determinant_via_multilinear_functionals [2014/03/21 11:11] (current) 2013/09/17 23:21 nikolaj 2013/09/17 23:20 nikolaj old revision restored (2013/09/17 23:06) Next revision Previous revision 2013/09/17 23:21 nikolaj 2013/09/17 23:20 nikolaj old revision restored (2013/09/17 23:06) Line 1: Line 1: - ===== Determinant via multilinear ​functional ​===== + ===== Determinant via multilinear ​functionals ​===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $V$ ... finite dimensional $\mathcal F$-vector space | + | @#55CCEE: context ​    | @#55CCEE: $V$ ... finite dimensional $\mathcal F$-vector space | - | @#FFBB00: $\mathrm{det}:​L(V,​V)\to \mathcal F$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\mathrm{det}:​L(V,​V)\to \mathcal F$ | - | @#DDDDDD: $n\equiv \mathrm{dim}(V)$ | + | @#DDDDDD: range       | @#DDDDDD: $n\equiv \mathrm{dim}(V)$ | | $M\in \mathrm{MultiLin}(V^n)$ | | $M\in \mathrm{MultiLin}(V^n)$ | | $v_1,​\dots,​v_n\in V$ | | $v_1,​\dots,​v_n\in V$ | | $A\in L(V,V)$ | | $A\in L(V,V)$ | - | @#55EE55: $M(A\ v_1,​\dots,​A\ v_n) = \mathrm{det}(A)\cdot M(v_1,​\dots,​v_n)$ | + | @#55EE55: postulate ​  | @#55EE55: $M(A\ v_1,​\dots,​A\ v_n) = \mathrm{det}(A)\cdot M(v_1,​\dots,​v_n)$ | ==== Discussion ==== ==== Discussion ==== Line 26: Line 26: Wikipedia: [[http://​en.wikipedia.org/​wiki/​Determinant|Determinant]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Determinant|Determinant]] ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Multilinear functional]],​ [[Linear operator algebra]], [[Vector space dimension]] [[Multilinear functional]],​ [[Linear operator algebra]], [[Vector space dimension]]