Basis for the vector space P2 I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial 
$
ax^2 + bx + c
$
with the matrix 
$
A = \begin{bmatrix}
1,0,0 \\
0,1,0 \\
0,0,1 \\
\end{bmatrix}
$
and the vector 
$
\begin{bmatrix}
1 \\
x \\
x^2 \\
\end{bmatrix}
$
what corresponds to $a$, $b$, and $c$ in the matrix $A$?

You shouldn't be representing the polynomial via the matrix $A$. Instead what you can think of is the following way of identifying a polynomial with a three dimensional vector.
$$ax^2+bx+c \Leftrightarrow \begin{bmatrix}a\\b\\c\end{bmatrix}$$
