Recharacterizing a function based on linearity assumption

Question

I know that a functional $L:$ $V$-->$W$ is linear if, for any vectors $u$,$v$ in $V$ and any scalars $a$,$b$ in $R$, $L(au+bv)=aL(u)+bL(v)$. The proof I am working on gives me the following function:

$f(x,y,z)=(2x-4y+3z+q, 6x + rxyz)$ where $q,r$ are in $R$. My task is to show that this function is linear if and only if $q=r=0$. I am first going in the "only if" direction, so I'm assuming linearity and trying to show that $r=q=0$.

My question is, how can I recharacterize this equation based on the fact that it's linear? I'm looking at the definition, but I'm not sure how to proceed when there are this many elements.

Answer 1

You need to apply the linearity assumption for well-chosen $u,v,a,b$.

Linearity implies $f(0,0,0)=0$. [Concretely, $L(0u + 0v) = 0 L(u) + 0 L(v) = 0$.] But $f(0,0,0) = (q, 0)$ so $q = 0$. To see that $r=0$, note that linearity implies $2 f(1, 1, 1) = f(2, 2, 2)$.