Let $T : \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be defined by $$T(x, y, z) = (2x + y, x − z, x + y^2).$$
Determine whether or not T is a linear transformation. If it is, justify. If not, show why.
I'm mainly just checking my work.
I concluded that $T$ is not a linear transformation. This is because when showing the vector additions are equal, the last term will not work out. This is because of the $y^2$ term.
Is this true?
Yes, the above is not a linear transformation, because of the $y^2$ sitting in the third term. The best way to bring this out is to exhibit a counterexample explicitly showing the issue. $T(0,1,0) = (1,0,1)$ and $T(0,-1,0) = (-1,0,1)$, so $T(0,1,0) + T(0,-1,0) = (0,0,2)$.
However, $T((0,1,0) + (0,-1,0)) = T((0,0,0)) = (0,0,0) \neq (0,0,2)$. Hence, $T$ is not linear.