# Finding Bases from polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? $$p_1 ( x ) = 4 x + 1$$ $$p_2 ( x ) = x^2 − 2 x + 3$$ $$p_3 ( x ) = 3 x − 2$$ $$p_4 ( x ) = x^2 − x + 5$$ (Hint: Use the standard basis for the space of second degree polynomials. Work with coordinate vectors written relative to this basis)

Can anyone put me on the right track ?

Hint: associate the second degree polynomial $ax^2+bx+c$ with the vector $\begin{bmatrix}a\\b\\c\end{bmatrix}$.
HINT: You must obtain $1$, $x$, $x^2$. First two of them are obtained as linear combinations of $p_1$ and $p_3$. Is it important which one of the remaining two you will take as the third vector?