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
Context
