Homomorphism space/Direct sum decomposition/Fact

{{ Mathematical text/Fact |Text= {{ Factstructure|typ= |Situation= Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ and ${\displaystyle {}W}$ be ${\displaystyle {}K}$-vector spaces. Let

${\displaystyle {}V=V_{1}\oplus \cdots \oplus V_{n}\,}$

and

${\displaystyle {}W=W_{1}\oplus \cdots \oplus W_{m}\,}$

de direct sum decompositions and let

${\displaystyle p_{j}\colon W\longrightarrow W_{j}}$

denote the canonical projections. |Condition= |Segue= |Conclusion= Then the mapping {{ Mapping/display |name= | \operatorname{Hom}_{ K } { \left( V , W \right) } | \prod_{1 \leq i \leq n,\, 1 \leq j \leq m } \operatorname{Hom}_{ K } { \left( V_i , W_j \right) } |f| p_j \circ {{mabr| f {{|}}_{V_i} |}} |pm=, }} is an isomorphism. |Extra=If we consider ${\displaystyle {}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)}}$ as linear subspaces of ${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}}$, then we have the direct sum decomposition

${\displaystyle {}\operatorname {Hom} _{K}{\left(V,W\right)}\cong \bigoplus _{1\leq i\leq n,\,1\leq j\leq m}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)}\,.}$

}} |Textform=Fact |Category= |Request=Direct sum decomposition of the space of homomorphisms }}