I'm supposed to show that a function is an inner product if and only if b^2 - ac < 0 and a > 0.
I have proven all of the properties except that <x,x> > 0 if x!= 0. I would write the function out, but can't seem to get matrices to work.
The function is the product of a 1x2 matrix with entries x1 and x2, a 2x2 matrix with entries a, b, b, c, (filling in the top row, then moving to the bottom row), and a 2x1 matrix with entries y1, y2.
After multiplication, I arrive at
<x,x> = a(x1)^2 + 2b(x1)(x2) + c(x2)^2 > 0.
Now I must prove that this is true if and only if b^2 - ac < 0 and a > 0, but am not sure how to go about this.
The Attempt at a Solution
I thought about different ways of inserting the conditions into the inequality or assuming that they were false and trying to arrive at a contradiction, but I can't seem to utilize them in a useful way.