Homomorphism space/Direct sum decomposition/Fact/Proof

Proof

It follows directly from fact that the given mapping is linear. In order to prove injectivity, let ${}f\in \operatorname {Hom} _{K}{\left(V,W\right)}$ with ${}f\neq 0$ be given. Then there exists some ${}v\in V$ such that

${}f(v)\neq 0\,.$

Let ${}v=v_{1}+\cdots +v_{n}$ with ${}v_{i}\in V_{i}$. Then also ${}f(v_{i})\neq 0$ for some ${}i$. Therefore, ${}(f(v_{i}))_{j}\neq 0$ for some ${}j$. Hence

${}f_{ij}=p_{j}\circ {\left(f{|}_{V_{i}}\right)}\neq 0\,.$

In order to prove surjectivity, let a family of homomorphisms ${}f_{ij}\in \operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)},\,1\leq i\leq n,\,1\leq j\leq m$, be given, which we consider as mappings to ${}W$. Then the

${}f_{i}=\sum _{j=1}^{m}f_{ij}\,$

are linear mappings from ${}V_{i}$ to ${}W$. This yields via fact a linear mapping ${}\sum _{i=1}^{n}f_{i}$ from ${}V$ to ${}W$, which restricts to the given mappings.

To fact