Let $\pi_n : R[x] \rightarrow R[x]_n$ be the map that cuts off all terms of higher than nth degree, i.e., $\pi_n(\sum_{i \geq 0} a_i x^i) = \sum_{i=0}^{n} a_i x^i$.
Let $q \in R[x]_m$ be a fixed polynomial and look at the figure
$M_q : R[x]_n \rightarrow R[x]_n$
$p \rightarrow \pi_n(p q)$
(a) Show that $M_q$ is a linear map
(b) Find the matrix of $M_q$ with respect to the standard basis $(1 , x, \ldots , x^n)$.
(c) For which $q$ is $M_q$ injective? Which polynomials $q$ lead to the same figure?
i have tried to prove the linearity in a)
to show $M_q(p_1 + p_2) = M_q(p_1) + M_q(p_2)$
For any $p_1, p_2 \in R[x]_n$ and q is m polynomial , we have
\begin{align*} M_q(c_1p_1 + c_2p_2) &= \pi_n\left(\sum_{i=0}^{n} c_1a_i x^i q + \sum_{i=0}^{n} c_2b_i x^i q\right) \\ &= \pi_n\left(\sum_{i=0}^{n}(c_1a_i + c_2b_i) x^i q\right) \\ &= \pi_n\left(\sum_{i=0}^{n+m}(c_1a_i + c_2b_i) x^i\right) \\ &= \left(\sum_{i=0}^{n}(c_1a_i + c_2b_i) x^i\right) \\ &= \left(c_1\sum_{i=0}^{n} a_i x^i + c_2\sum_{i=0}^{n} b_i x^i\right) \\ \end{align*}
after this i'm not sure, like i have got polynomial degree n now, so i just restore $\pi_n$ where we got sum start from i bigger or equal 0 and goes up, so i can factorize q since i can have the sum bigger than n in it ? and just like this finish proving the linearity with the missing 2 or 3 steps ? and $\pi_n$ will it return any thing to grade n? i.e for example $\pi_n(x)$ will give me polynomial of grade n ?
regarding b)
To find the matrix of $(M_q)$ with respect to the standard basis $((1, x, ...., x^{n}))$, we need to apply $(M_q)$ to each basis vector and express the results in terms of the standard basis vectors
\begin{align*} 1.& \quad M_q(1) = \pi_n(q) = (q_0, q_1, \ldots, q_n)\\ 2.& \quad M_q(x) = \pi_n(xq) = (\pi_n(0), \pi_n(q_0), \pi_n(q_1), \ldots, \pi_n(q_{n-1})) = (0, q_0, q_1, \ldots, q_{n-1})\\ 3.& \quad M_q(x^2) = \pi_n(x^2q) = (\pi_n(0), \pi_n(0), \pi_n(q_0), \ldots, \pi_n(q_{n-2})) = (0, 0, q_0, \ldots, q_{n-2})\\ 4.& \quad \vdots \\ 5.& \quad M_q(x^n) = \pi_n(x^nq) = (\pi_n(0), \ldots, \pi_n(0), q_0) = (0, \ldots, 0, q_0) \end{align*}
regarding c) i would say q=0 to ensure that $M_q$(p)=0 for p any grade chosen, and like this we ensure the injectivity
is this the correct way ?