linear_operator

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

Left module homomorphism