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linear_operator_space [2014/03/21 11:11]
127.0.0.1 external edit
linear_operator_space [2015/10/29 12:55] (current)
nikolaj
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 ==== Set ==== ==== Set ====
 | @#55CCEE: context ​    | @#55CCEE: $X,​Y$...left $\mathcal R$-module | | @#55CCEE: context ​    | @#55CCEE: $X,​Y$...left $\mathcal R$-module |
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 | @#FFBB00: definiendum | @#FFBB00: $\langle\mathrm{Hom}(X,​Y),​+,​\cdot \rangle \in \mathcal L(X,Y)$ | | @#FFBB00: definiendum | @#FFBB00: $\langle\mathrm{Hom}(X,​Y),​+,​\cdot \rangle \in \mathcal L(X,Y)$ |
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 | @#55CCEE: context ​    | @#55CCEE: $+:​\mathrm{Hom}(X,​Y)\times \mathrm{Hom}(X,​Y)\to \mathrm{Hom}(M,​N)$ | | @#55CCEE: context ​    | @#55CCEE: $+:​\mathrm{Hom}(X,​Y)\times \mathrm{Hom}(X,​Y)\to \mathrm{Hom}(M,​N)$ |
 | @#55CCEE: context ​    | @#55CCEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,​Y)\to\mathrm{Hom}(X,​Y)$ | | @#55CCEE: context ​    | @#55CCEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,​Y)\to\mathrm{Hom}(X,​Y)$ |
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 | $ v\in M $ | | $ v\in M $ |
 | $r,s \in \mathcal R$ | | $r,s \in \mathcal R$ |
 | $A,B \in \mathrm{Hom}(X,​Y)$ | | $A,B \in \mathrm{Hom}(X,​Y)$ |
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 | @#55EE55: postulate ​  | @#55EE55: $(r \cdot  A+s \cdot  B)\ v = r\ (A\ v) + s\ (B\ v) $ | | @#55EE55: postulate ​  | @#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 ====+ 
 +-----
 === Context === === Context ===
 [[Left module homomorphism]] [[Left module homomorphism]]
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