Prove that T is a linear transformation Does it matter that in the first line it's written $T(\alpha p+ \beta g)$ and not $T(\alpha p(t)+ \beta g(t))$ but at the end it is written with $\alpha T(p(t)) + \beta T(g(t)))$ with the $t$'s.

Define $T : \mathbb{P}_3 \to \mathbb{R}^4$ by $$T(p) = \begin{bmatrix}
p(-3) \\
p(-1) \\
p(1) \\
p(3)
\end{bmatrix}$$ Show that $T$ is a linear transformation. 



It doesn't really matter. But it doesn't look very good. I would've written $\alpha T(p) + \beta T(g)$ the last time. Also, why choose the letters $p$ and $g$? Why not $f$ and $g$, or $p$ and $q$? That would make the proof look nicer, at least to my eyes.
