Homomorphism space/Direct sum decomposition/Fact/Proof

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

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

Let ${\displaystyle {}v=v_{1}+\cdots +v_{n}}$ with ${\displaystyle {}v_{i}\in V_{i}}$. Then also ${\displaystyle {}f(v_{i})\neq 0}$ for some ${\displaystyle {}i}$. Therefore, ${\displaystyle {}(f(v_{i}))_{j}\neq 0}$ for some ${\displaystyle {}j}$. Hence {{ Relationchain/display | f_{ij} || p_j \circ {{mabr| f {{|}}_{V_i} |}} |\neq|0 || || |pm=. }}

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

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

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