Consider vector space $P_2$, calculate components of the polynomial $p(x)=-3x^2+7x+5 $ in the basis $C=[x+1,3x+2,x^2+x+5]$.
My working:
We get matrix $\begin{pmatrix} 1&2&5 \\ 1&3&1 \\ 0&0&1 \end{pmatrix}$ directly from the basis, C.
Then, we find the inverse of this matrix, to find the change of basis matrix: $\begin{pmatrix} 1&2&5 \\ 1&3&1 \\ 0&0&1 \end{pmatrix}^{-1} = \begin{pmatrix} 3&-2&-13 \\ -1&1&4 \\ 0&0&1 \end{pmatrix}$
Then, we can right multiple the coefficients of the polynomial we need to find in the new basis with the change of basis matrix we found.
i.e. $\begin{pmatrix} 3&-2&-13 \\ -1&1&4 \\ 0&0&1 \end{pmatrix}\begin{pmatrix} 5\\7\\-3 \end{pmatrix}=\begin{pmatrix} 40 \\ -10 \\ -3 \end{pmatrix}$
Hence we have $p(x)=40x^2-10x-3 $ in the new basis C
Was just unsure as I have not covered this in a while.