I'm doing a bit of homework and want to check if I'm doing it right. The question is as follows:
Let $T: P_2(\mathbb{R}) \rightarrow P_3(\mathbb{R})$ be defined by $T(p)(x) = (x-1)p(x)$
We're then asked to compute the matrix of $T$ with respect to the standard basis $\{1, x, x^2\}$. I go through that and get:
$A = \begin{bmatrix}-1&0&0\\ 1&-1&0\\0&1&-1\\0&0&1\end{bmatrix}$
Is this correct? I think it is but I'm not really sure.
However my main question is regarding $range T$. How would one describe the range of $T$ without using the matrix? I tried doing it like
Let $p \in P_3({\mathbb{R})} $ then
$T(p)(x) = (x-1)(a + bx + cx^2)$ for $a,b,c, \in \mathbb{R}$
$T(p)(x) = ax + bx^2 + cx^3 -a -bx -cx^2 = (a-b)x + (b-c)x^2 + cx^3 - a$
So, the range of T would be $\{ (a-b)x + (b-c)x^2 + cx^3 - a : a,b,c \in \mathbb{R}\}$?