I have these maps from the plane to itself where $X=(x,y)$:
$f(X):=(y,-x)$
$g(X):=(x+2y,y)$
I need to compute $fg$ and $gf$ and show that none of these compositions are simply translations or that for a point $(x,y)$, it shouldn't be mapped to $(x+a_1,y+a_2)$ for some $a_1,a_2 \in R$.
For the first composition, namely $fg$, I tried plugging in $g$ to $(x,y)$ giving $(x+2y,y)$. Then plugging these into $f$ giving $(y,-x-2y)$.
Now is this correct and enough to say that this isn't simply a translation given by the fact that x and y appear in the second term? And then could I apply this technique to the other composition, namely $gf$, for the other portion of the problem?