Homomorphism space/Direct sum decomposition/Fact

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

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

and

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

de direct sum decompositions and let

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

denote the canonical projections.

Then the mapping

$\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \prod _{1\leq i\leq n,\,1\leq j\leq m}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)},f\longmapsto p_{j}\circ {\left(f{|}_{V_{i}}\right)},$

is an

isomorphism.

If we consider ${}\operatorname {Hom} _{K}{\left(V_{i},W_{j}\right)}$ as linear subspaces of ${}\operatorname {Hom} _{K}{\left(V,W\right)}$, then we have the direct sum decomposition

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

Proof, Write another proof