i'm having trouble figuring out how to approach this problem. i've been able to solve similar problems involving vectors, but this one involves polynomials and i can't find a useful example for how to deal with it.
$A$ and $B$ are pairs of ordered bases for second degree polynomials. i need to find the change of coordinate matrix from $A$ coordinates to $B$ coordinates.
for the first question, $A = \{1, x, x^2\}$ and $B = \{2x^2-x, 3x^2+1, x^2\}$
i know i'm supposed to express each element of $A$ as a linear combination of all elements in $B$, e.g.,
$1 = a(2x^2-x)+b(3x^2+1)+c(x^2)$
but i can't remember how to set this up as a matrix. i know i'm supposed to collect coefficients and then arrange them in rows according to the order of their corresponding variables. e.g.,
$1 = (2a+3b+c)x^2+(-a)x+(b)$
i think i need a row for the coefficients of the $x^2$ term, the $x$ term and the constant term, but i don't know what to set them equal to here.
what should i do? thanks in advance.
Now, solve the system:$$\left\{\begin{array}{l}2a+3b+c=0\\-a=0\\b=1.\end{array}\right.$$The solution will be $a=-\frac32$, $b=1$, and $c=0$. In other words, the first element of $A$ is $-\frac32\times(2x^2-x)+1\times(3x^2+1)+0\times x^2$. So, the entries of the first column of your matrix will be $-\frac32$, $1$, and $0$.
Now, do the same thing with the second and the third elements of $A$.