===== Linear operator space =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $X,Y$...left $\mathcal R$-module |
| @#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: $\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ |
| $ v\in M $ |
| $r,s \in \mathcal R$ |
| $A,B \in \mathrm{Hom}(X,Y)$ |
| @#55EE55: postulate   | @#55EE55: $(r \cdot  A+s \cdot  B)\ v = r\ (A\ v) + s\ (B\ v) $ |

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=== 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

$A\ e_i=\sum_{j=1}^n A_{i,j}\cdot e_j$
=== Reference ===
Wikipedia: [[https://en.wikipedia.org/wiki/Left_module#Submodules_and_homomorphisms|Module]]

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=== Context ===
[[Left module homomorphism]]