How do you find a basis for a polynomial in P2 given a set of polynomials? I don't know how to show that p1, p2, and p3 actually form a basis for P2. I have been trying different things, but that fixed scalar c has prevented me from forming a basis.
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 A: Hint: Start by assuming
$$a_1p_1 + a_2p_2 + a_3p_3 = 0$$
for some scalars $a_1, a_2, a_3$.  You have to show that in fact $a, b, c$ are all zero.  To do this remember that the equation above is an equality of polynomials.  So the coefficients of $x^2$ on the left should equal the coefficient of $x^2$ on the right (which is zero!).  Same for the coefficient of $x$ and for the constant term.
That shows that the set is linearly independent.  Now if you know that the dimension of $P_2$ is $3$ then you're done.  If not then you'll need to also show this set spans.  So start with the equation
$$a_1p_1 + a_2p_2 + a_3p_3 = c_2x^2 + c_1x + c_0$$
and equate coefficients like you did above.  This gives you a system of equations where $c, c_0, c_1, c_2$ are treated just like numbers and $a_1, a_2, a_3$ are variables.  You need to show that this system of equations has a solution.
Note, if you're extra slick you can just do the second step and then use the fact that the solution you find is unique in order to do the first step automatically.
