<s>[INST] Change of polynomial basis exercise Can someone please check if my solution is correct? Thanks.
Considering the following basis:
$$
B= \left\{ x^2,1+x^2,x-1 \right\}\\
C= \left\{ x,1-x,x^2 \right\}
$$
a) Find the change of basis matrices from $B$ to $C$ and $C$ to $B$.
b) Find the coordinates of the vector $x+1$ in $C$.
c) Find a basis $D$ of $R^2[x]$ such that the change of basis matrix from $D$ to $C$ is the same as the change from $C$ to $B$.
My Solution:
a)To reach what I've done, I decomposed the vectors in one basis in terms of a linear combination of the spam of the other basis...
From $B$ to $C$:
$$
\begin{pmatrix}
0 & 1 & 0\\
0 & 1 & -1\\
1 & 1 & 0\\
\end{pmatrix}
$$
From $C$ to $B$:
$$
\begin{pmatrix}
-1 & 0 & 1\\
1 & 0 & 0\\
1 & -1 & 0\\
\end{pmatrix}
$$
b) x+1 in $C$:
$$
\alpha x + \beta (1-x) + \gamma x^2 = x+1\\
\gamma = 0\\
\alpha - \beta = 1\\
\beta = 1\\
\left[\begin{matrix}
2\\
1\\
0
\end{matrix}\right] 
$$
c)Basis $D$:
First vector is going to be the first column of this $C$ to $B$ matrix, times the vectors in basis $C$:
$$
\left[\begin{matrix}
-1\\
1\\
1\\
\end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ =
-x+1-x+x^2 = \\
x^2-2x+1
$$
Same for second and third vectors:
$$
\left[\begin{matrix}
0\\
0\\
-1\\
\end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ = -x^2\\
\left[\begin{matrix}
1\\
0\\
0\\
\end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ = x\\
$$
So the basis $D$ is going to be:
$$
D= \left\{ x^2-2x+1,-x^2,x \right\}
$$
Is everything correct?!
Thanks.
 [/INST] Everything looks correct. The only thing I would say  is that you should make sure D is in fact a basis by checking the vectors are linearly independent and checking  whether they in fact span $R^2[x]$. This should be fairly easy by looking at the dimension of the subspace or the linear independence of the coordinates.
</s>