Suggest edit — Product of Linear Maps

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---
title: "Product of Linear Maps"
type: concept
related: [Identity, Linear Map]
source: https://www.jemoka.com/posts/kbhproduct_of_linear_maps/
confidence: high
status: active
---

Take two linear maps \(T \in \mathcal{L}(U,V)\) and \(S \in \mathcal{L}(V,W)\), then \(ST \in \mathcal{L}(U,W)\) is defined by:
\begin{equation} (ST)(u) = S(Tu) \end{equation}
Indeed the &ldquo;product&rdquo; of Linear Maps is just function composition. Of course, \(ST\) is defined only when \(T\) maps to something in the domain of \(S\).
The following there properties hold on linear-map products (note that commutativity isn&rsquo;t one of them!):
associativity \begin{equation} (T_1T_2)T_3 = T_1(T_2T_3) \end{equation}
identity \begin{equation} TI = IT = T \end{equation}
for \(T \in \mathcal{L}(V,W)\) and \(I \in \mathcal{L}(V,V)\) (OR \(I \in \mathcal{L}(W,W)\) depending on the order) is the identity map in \(V\).
identity commutes, as always.
distributive in both directions&mdash;
\begin{equation} (S_1+S_2)T = S_1T + S_2T \end{equation}
and
\begin{equation} S(T_1+T_2) = ST_{1}+ST_{2} \end{equation}