<s>[INST] The map $p(x)\mapsto p(x+1)$ in vector space of polynomials I encountered this question in a previous year paper of an exam. I wish somebody could help me how to go by this question.
Choose the correct option
Let $N$ be the vector space of all real polynomials of degree at most $3$. Define
$$S:N \to N \ \text{by} \ (Sp)(x)=p(x+1), \ p \in N$$
Then the matrix of $S$ in the basis $\{1,x,x^2,x^3\}$, considered as column vectors, is given by


*

*\begin{bmatrix}1 &0 &0& 0\\
  0& 2& 0& 0\\
0 &0& 3& 0\\
0& 0& 0& 4\end{bmatrix}

*\begin{bmatrix}1& 1& 1& 1\\
0& 1 &2 &3\\
0 &0& 1& 3\\
0& 0 &0 &1\end{bmatrix}

*\begin{bmatrix}1& 1& 2& 3\\
1 &1 &2& 3\\
2 &2 &2 &3\\
3 &3& 3 &3\end{bmatrix}

*\begin{bmatrix} 0& 0& 0& 0\\
1& 0 &0 &0\\
0& 1& 0& 0\\
0 &0 &1& 0\end{bmatrix}


How to tackle p(x+1) thing?
 [/INST] So, you can begin computing the images of $1, x, x^2$ and $x^3$. For instance:
$$
S(1) = 1 \ ,
$$
because it doesn't matter wherever you evaluate the constant polynomial $1$, its value will always be $1$. Next, you compute also
$$
S(x) = x + 1 \ , \qquad S(x^2) = (x + 1)^2 = x^2 + 2x + 1 
$$
and
$$
\qquad S(x^3) = (x+1)^3 = \dots
$$
Then, you think about the coordinates of these polynomials in the basis $1, x, x^2, x^3$. For instance,
$$
1 = 1\cdot 1 + 0\cdot x +0 \cdot x^2 + 0 \cdot x^3 \ .
$$
So the coordinates of $S(1) = 1$ are
$$
\begin{pmatrix}
1 \\
0  \\
0 \\
0 \\
\end{pmatrix}
$$
This way you've obtained the first column of your matrix. The second column will be the coordinates of $S(x) = x + 1$. That is,
$$
\begin{pmatrix}
1 \\
1  \\
0 \\
0 \\
\end{pmatrix}
$$
The third one, those of $S(x^2)$,
$$
\begin{pmatrix}
1 \\
2  \\
1 \\
0 \\
\end{pmatrix}
$$
And the last one?
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