Q:

Transformation between vector space that removes scalar coefficient

I'm currently studying up on linear transformations and was wondering if you can define a transformation between two vector spaces like so (a is a scalar constant and x is a vector):
$T(ax) = x$
I'm not sure if this means anything, since $ax$ is itself another vector, so how would you be able to "remove" the $a$ from an arbitrary input vector? 

A:

Your intuition is correct, you may not define such a thing in general and here is why: if $x = a y$, then $T(x) = T(ay) = T(y)$; this shows that the only such $T$ are the ones that may be transported downto the projective space associated to your vector space - in particular, such transformations are not linear.  If projective spaces are foreign to you, rephrase the above as "the only such transformations are the ones that are constant along $1$-dimensional subspaces (i.e. straight lines)".