# Differences

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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 Both sides previous revision Previous revision 2015/10/29 12:55 nikolaj 2014/03/21 11:11 external edit2013/09/17 13:46 nikolaj 2013/09/13 21:45 nikolaj 2013/09/13 21:41 nikolaj created2013/09/13 21:38 nikolaj removed2013/09/06 22:04 external edit2013/08/31 19:56 nikolaj 2013/08/31 19:54 nikolaj 2013/08/31 19:54 nikolaj 2013/08/31 17:57 nikolaj 2013/08/31 17:51 nikolaj 2013/08/31 17:51 nikolaj 2013/08/31 17:47 nikolaj created 2015/10/29 12:55 nikolaj 2014/03/21 11:11 external edit2013/09/17 13:46 nikolaj 2013/09/13 21:45 nikolaj 2013/09/13 21:41 nikolaj created2013/09/13 21:38 nikolaj removed2013/09/06 22:04 external edit2013/08/31 19:56 nikolaj 2013/08/31 19:54 nikolaj 2013/08/31 19:54 nikolaj 2013/08/31 17:57 nikolaj 2013/08/31 17:51 nikolaj 2013/08/31 17:51 nikolaj 2013/08/31 17:47 nikolaj created Line 2: Line 2: ==== Set ==== ==== Set ==== | @#55CCEE: context ​    | @#55CCEE: $X,​Y$...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: definiendum | @#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: $+:​\mathrm{Hom}(X,​Y)\times \mathrm{Hom}(X,​Y)\to \mathrm{Hom}(M,​N)$ | | @#55CCEE: context ​    | @#55CCEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,​Y)\to\mathrm{Hom}(X,​Y)$ | | @#55CCEE: context ​    | @#55CCEE: $\cdot : \mathcal R\times\mathrm{Hom}(X,​Y)\to\mathrm{Hom}(X,​Y)$ | - | $v\in M$ | | $v\in M$ | | $r,s \in \mathcal R$ | | $r,s \in \mathcal R$ | | $A,B \in \mathrm{Hom}(X,​Y)$ | | $A,B \in \mathrm{Hom}(X,​Y)$ | - | @#55EE55: postulate ​  | @#55EE55: $(r \cdot A+s \cdot B)\ v = r\ (A\ v) + s\ (B\ v)$ | | @#55EE55: 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 Line 20: Line 17: === 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]]