Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | |||
linear_operator_space [2014/03/21 11:11] 127.0.0.1 external edit |
linear_operator_space [2015/10/29 12:55] nikolaj |
||
---|---|---|---|
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]] |