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linear_operator_space [2013/09/17 13:46] nikolaj |
linear_operator_space [2015/10/29 12:55] nikolaj |
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| ===== Linear operator space ===== | ===== Linear operator space ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X,Y$...left $\mathcal R$-module | | + | | @#55CCEE: context | @#55CCEE: $X,Y$...left $\mathcal R$-module | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $\langle\mathrm{Hom}(X,Y),+,\cdot \rangle \in \mathcal L(X,Y)$ | | |
| - | | @#FFBB00: $\langle\mathrm{Hom}(X,Y),+,\cdot \rangle \in \mathcal L(X,Y)$ | | + | | @#55CCEE: context | @#55CCEE: $+:\mathrm{Hom}(X,Y)\times \mathrm{Hom}(X,Y)\to \mathrm{Hom}(M,N)$ | |
| - | + | | @#55CCEE: context | @#55CCEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ | | |
| - | | @#88DDEE: $+:\mathrm{Hom}(X,Y)\times \mathrm{Hom}(X,Y)\to \mathrm{Hom}(M,N)$ | | + | |
| - | | @#88DDEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ | | + | |
| | $ v\in M $ | | | $ v\in M $ | | ||
| | $r,s \in \mathcal R$ | | | $r,s \in \mathcal R$ | | ||
| | $A,B \in \mathrm{Hom}(X,Y)$ | | | $A,B \in \mathrm{Hom}(X,Y)$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $(r \cdot A+s \cdot B)\ v = r\ (A\ v) + s\ (B\ v) $ | | ||
| - | | @#55EE55: $(r \cdot A+s \cdot B)\ v = r\ (A\ v) + s\ (B\ v) $ | | + | ----- |
| - | + | === Discussion === | |
| - | ==== Discussion ==== | + | |
| A linear operator $A:X\to X$ over an $n$-dimensional vector space can be encoded in a [[matrix]] and if $\{v_1,\dots,v_n\}$ is a basis then for all $1\ge i \ge n$ one has | A linear operator $A:X\to X$ over an $n$-dimensional vector space can be encoded in a [[matrix]] and if $\{v_1,\dots,v_n\}$ is a basis then for all $1\ge i \ge n$ one has | ||
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| === Reference === | === Reference === | ||
| Wikipedia: [[https://en.wikipedia.org/wiki/Left_module#Submodules_and_homomorphisms|Module]] | Wikipedia: [[https://en.wikipedia.org/wiki/Left_module#Submodules_and_homomorphisms|Module]] | ||
| - | ==== Parents ==== | + | |
| - | === Requirements === | + | ----- |
| + | === Context === | ||
| [[Left module homomorphism]] | [[Left module homomorphism]] | ||
