<s>[INST] If $B= ((X-1)^2, X-1, 1)$ is a base of $\mathbb{R}_2[X]$, how do I the coordinates of a polynomial in this base? Suppose you have a base: $B_2=((X-1)^2, (X-1), 1)$ and a polynomial of the form $a + bX + cX^2$ in base $B_1(X^2, X, 1)$, how do I find the its coordinates in $B_2$?
I figured maybe trying to find the matrix of the base change, and as $B_2=((X^2 -2X +1), (X-1),1))$ this would give: $$ \begin{bmatrix}
1 & 0 &0 \\ 
-2&  1& 0\\ 
1&  -1& 1
\end{bmatrix}$$
And the coordinates of the polynomial in $B_1$ are $(a,b,c)$, then I would simply multiply the vector of the coordinates by the $B_2$, which yields this $$[a-2b+c, b-c, c ] $$
Is this correct?
 [/INST] \begin{align} 
x^2 &= 1\cdot(x-1)^2 + 2\cdot(x-1) + 1\cdot1 \\
x &= 1\cdot(x-1) + 1\cdot1 \\ 
1 &= 1\cdot1 \\ \\
\Rightarrow ax^2 + bx + c &= a((x-1)^2 + 2(x-1) + 1) + b((x-1) + 1) + c \\
&= a\cdot(x-1)^2 + (2a + b)\cdot(x-1) + (a+b+c)\cdot 1
\end{align}
That is we need something that maps $(a,b,c)$ to $(a, 2a+b, a+b+c)$.
The matrix that belongs to this transformation is the one on the left side of the following equation.
$$\pmatrix{1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & 1 & 1} \pmatrix{a\\b\\c} = \pmatrix{a\\ 2a + b \\ a+  b+c}$$
</s>