If I have a subspace, S = {$p(x)$ $\in$ $P_2(R)$ | $p(1) = 0$} of $P_2(R)$.
$A$ = {$-1 + x, -1 + x^2$ } and $B$ = {$1 + x - 2x^2$ , $3 - 4x + x^2$ } are both bases for S.
What is the change of the coordinates matrices from $B$ to $A$ and $A$ to $B$?
i.e $_AP_B$ and what is $_BP_A$
I started to find the A-coordinates of vectors in B by doing
$1 + x - 2x^2$ = $c_1$($-1 + x)$ + $c_2$($-1 + x^2$) = $ -1 (c_1 +c_2) + (c_1) x + (c_2) x^2$
$3 - 4x + x^2$ = $d_1$($-1 + x)$ + $d_2$($-1 + x^2$) = $ -1 (d_1 +d_2) + (d_1) x + (d_2) x^2$
; but was unsure how to continue and form this into an augmented matrix
ETA
So my augmented matrix will end up being:
$ \left[ \begin{array}{ccc|cc} 1 & 1 & -2 & 1 & -2 \\ 3 & -4 & 1 & 1 & -4 \\ \end{array} \right] $
for the A-coordinates of vectors in B?
Please edit the matrix to correct it if/because it is wrong.