Direct sum/Linear mapping/Fact

Let ${}V$ be a ${}K$-vector space together with a direct sum decomposition

${}V=U_{1}\oplus U_{2}\,.$

Let ${}W$ be another ${}K$-vector space and let

$\varphi _{1}\colon U_{1}\longrightarrow W$

and

$\varphi _{2}\colon U_{2}\longrightarrow W$

denote linear mappings.

Then we get, by setting

${}\varphi (v):=\varphi _{1}(v_{1})+\varphi _{2}(v_{2})\,,$

where ${}v=v_{1}+v_{2}$ is the direct decomposition, a linear mapping

$\varphi \colon V\longrightarrow W.$

Proof, Write another proof