Map not preserving vector addition but preserving scalar multiplication The question
Map closed under addition but not multiplication
asks for a map between two vector spaces where vector addition is preserved but also where scalar multiplication is not preserved.
A student of mine switched this, and I have not found an example.
Thus, what is an example of a map between two vector spaces where vector addition is not preserved but scalar multiplication is preserved?
Thank you.
 A: Unlike the reverse question, in this case the condition is weak enough that we can find examples even over $\mathbb R$. Essentially, this is telling us that the map acts like a linear map on every line through the origin, but isn't consistent about what it does to different lines. (So we'll need at least a two-dimensional vector space to find a counterexample.)
One possible example is the map $T : \mathbb R^2 \to \mathbb R^2$ defined in polar coordinates as $$T\begin{bmatrix}r \cos \theta \\ r \sin \theta\end{bmatrix} = \frac1{|\cos \theta| + |\sin \theta|} \begin{bmatrix} r \cos \theta \\ r \sin \theta \end{bmatrix}$$ which bends the unit circle into a diamond. In general, you can distort the unit circle however you like (that's symmetric with respect to reflection through the origin), and then make sure that scalar multiplication is respected. (This corresponds to multiplying by $f(\theta)$ for any function $f : [0, 2\pi] \to \mathbb R$ such that $f(\theta + \pi) = f(\theta)$.)
A: How about $L:\mathbb R^2\to \mathbb R$ defined by:
$$  L(x,y) =
\begin{cases}
x,  & \text{if $y=0$ } \\
y, & \text{if $y\neq 0$ }
\end{cases}$$
A: I here share another example:
$T: \mathcal A\to \mathbb R$ defined by:
where $\mathcal A=\{[x,y]\in{\mathbb R}^2\mid\neg (x=0~\cap~y=0)\}$
$T(x,y) =
\begin{cases}
\cfrac{xy^2}{x^2+y^2}
\end{cases}$
