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Linear operator space
Definition
$X,Y$…left $\mathcal R$-module |
$\langle\mathrm{Hom}(X,Y),+,\cdot \rangle \in \mathcal L(X,Y)$ |
$+:\mathrm{Hom}(X,Y)\times \mathrm{Hom}(X,Y)\to \mathrm{Hom}(M,N)$ |
$\cdot : \mathcal R\times\mathrm{Hom}(X,Y)\to\mathrm{Hom}(X,Y)$ |
$ v\in M $ |
$r,s \in \mathcal R$ |
$A,B \in \mathrm{Hom}(X,Y)$ |
$(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
Parents
Requirements