TITLE: non linear transformation that satisfies $T(cx) = cT(x)$
QUESTION [5 upvotes]: I am just curious if there is a transformation that does not satisfy 
$\;T(x+y) = T(x) + T(y),\;$
 but satisfies $\;T(cx)=cT(x).\;$ 
I cannot think of any. Thanks for any help people.

REPLY [2 votes]: How about $T\colon\mathbb{R}^2\to\mathbb{R}^2$ by:
$$T(v)=\begin{cases}v,&\text{$v=(\lambda,0)$ for $\lambda\in\mathbb{R}$}\\0,&\text{else}\end{cases}$$
Writing $V$ for the span of $(1,0)$, we have $v\in V$ iff $cv\in V$ for every $c\in\mathbb{R}$, so $T(cv)=cT(v)$ (both sides $0$ if $v\notin V$ and both sides $cv$ if $v\in V$. However, $$T((1,0)+(0,1))=T(1,1)=0\ne (1,0)=T(1,0)+T(0,1).$$
Edit: This is a special case of Christoph's construction, with $\varphi(v)=1$ if $v\in V$ and $\varphi(v)=0$ otherwise.