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.
Best Answer
How about $L:\mathbb R^2\to \mathbb R$ defined by:
$$ L(x,y) = \begin{cases} x, & \text{if $y=0$ } \\ y, & \text{if $y\neq 0$ } \end{cases}$$