Direct sum/Linear mapping/Fact/Proof

Proof

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

\begin{matrix} φ (a v + b v^{'}) & = φ_{1} ((a v + b v^{'})_{1}) + φ_{2} ((a v + b v^{'})_{2}) \\ = φ_{1} (a v_{1} + b v'_{1}) + φ_{2} (a v_{2} + b v'_{2}) \\ = a φ_{1} (v_{1}) + b φ_{1} (v'_{1}) + a φ_{2} (v_{2}) + b φ_{2} (v'_{2}) \\ = a φ_{1} (v_{1}) + a φ_{2} (v_{2}) + b φ_{1} (v'_{1}) + b φ_{2} (v'_{2}) \\ = a φ (v) + b φ (v^{'}) \end{matrix}

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