**Let $f: \mathbb R^3 \longrightarrow R^3$ be given by
$$f:\left(\begin{array}{c}{x\_{1}} \\ {x\_{2}} \\ {x\_{3}}\end{array}\right) \longmapsto \left(\begin{array}{c}{x\_{1}+x\_{2}} \\ {x\_{2}} \\ {x\_{3}-x\_{1}}\end{array}\right)$$
Prove that $f$ is linear.**

*my answer*

I could use 

\begin{array}{l}{f(\mathbf{u}+\mathbf{v})=f(\mathbf{u})+f(\mathbf{v})} \\ {f(c \mathbf{u})=c f(\mathbf{u})}\end{array}

to prove that $f $ is linear. Put I have found a trick to do this without much effort. 

*Claim: $f$ is linear if the coordinates of $f(\bf u)$ are linear.* 

In this case $x\_1+x\_2$, $x\_2$ and $x\_3-x\_1$ are linear, therefore $f$ is linear.

Does this theorem and the belonging proof exist?