Direct sum/Linear mapping/Fact/Proof

The mapping is well-defined, since the representation ${\displaystyle {}v=v_{1}+v_{2}}$ with ${\displaystyle {}v_{1}\in U_{1}}$ and ${\displaystyle {}v_{2}\in U_{2}}$ is unique. The linearity follows from

${\displaystyle {}{\begin{aligned}\varphi (av+bv')&=\varphi _{1}{\left((av+bv')_{1}\right)}+\varphi _{2}{\left((av+bv')_{2}\right)}\\&=\varphi _{1}{\left(av_{1}+bv'_{1}\right)}+\varphi _{2}{\left(av_{2}+bv'_{2}\right)}\\&=a\varphi _{1}(v_{1})+b\varphi _{1}(v'_{1})+a\varphi _{2}(v_{2})+b\varphi _{2}(v'_{2})\\&=a\varphi _{1}(v_{1})+a\varphi _{2}(v_{2})+b\varphi _{1}(v'_{1})+b\varphi _{2}(v'_{2})\\&=a\varphi (v)+b\varphi (v')\end{aligned}}}$