Let $\mathbb{R}_3[x]$ be a vector space of polynomials p with degree $\leq3$ and show that $\phi: \mathbb{R}_3[x]\rightarrow\mathbb{R}^3, \phi(p):=[p(-1), p(0), p(1)] $ is a linear transformation.
Now I know that transformation is linear if these two conditions are true:
A linear transformation between two vector spaces $V$ and $W$ is a map $T:V\rightarrow W$ such that the following hold:
$T(v_1+v_2)=T(v_1)+T(v_2)$ for any vectors $v_1$ and $v_2$ in $V$, and
$T(\alpha v)=\alpha T(v)$ for any scalar $\alpha$.
Let $p(x)=\alpha p_1(x)+\beta p_2(x)$, now we have $\phi(p(x)) = \phi(\alpha p_1(x)+\beta p_2(x))=\left[\begin{array}{c}\alpha p_1(-1)+\beta p_2(-1)\\\alpha p_1(0)+\beta p_2(0)\\\alpha p_1(1)+\beta p_2(1)\end{array}\right]=\alpha\left[\begin{array}{c} p_1(-1)\\p_1(0)\\p_1(1)\end{array}\right]+\beta\left[\begin{array}{c} p_2(-1)\\p_2(0)\\p_2(1)\end{array}\right]$
Does this prove that the transformation is linear?