## Linear operator space

### Set

 context $X,Y$…left $\mathcal R$-module definiendum $\langle\mathrm{Hom}(X,Y),+,\cdot \rangle \in \mathcal L(X,Y)$ context $+:\mathrm{Hom}(X,Y)\times \mathrm{Hom}(X,Y)\to \mathrm{Hom}(M,N)$ context $\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ $v\in M$ $r,s \in \mathcal R$ $A,B \in \mathrm{Hom}(X,Y)$ postulate $(r \cdot A+s \cdot B)\ v = r\ (A\ v) + s\ (B\ v)$

#### 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: Module