i am considering two bases. i need to find the change of coordinate matrix that changes $B'$ coordinates into $B$ coordinates.
my first question had these two bases.
$B=\{x^2,x,1\}, B'=\{a_2x^2+a_1x+a_0,b_2x^2+b_1x+b_0,c_2x^2+c_1x+c_0\}$
my answer was $ \begin{pmatrix} a_2 & b_2 & c_2 \\ a_1 & b_1 & c_1 \\ a_0 & b_0 & c_0 \\ \end{pmatrix} $ and it was correct.
in the second problem, $B'$ was the same as before, but now $B=\{1,x,x^2\}$.
i got the same answer as before because when i turn each element of $B'$ into a linear combination of this definition of $B$, i think that the coefficients of the linear combination are the same as the ones for each element of $B'$. i.e., $a_2x^2+a_1x+a_0=ax^2+bx+c\mapsto a_2=a,a_1=b,a_0=c$.
but the answer for this question is the same as above with the first and third rows switched. (i.e., $a_2$ row on the bottom, $a_0$ row on the top. the answer i got is incorrect.)
how does the order of $B$ affect this matrix?
thank you in advance for any insight.
It affects because you have to write, say, the first element of $B'$ as$$\alpha\times\text{(first element of $B$)}+\beta\times\text{(second element of $B$)}+\gamma\times\text{(third element of $B$)}.$$So, if you exchange the order the answer changes too.