Q2 (12 points)
The following do not define inner products on the indicated vector space. Explain why.
a) V = \mathbb{R}^2, \langle x, y \rangle = x_1x_2y_1y_2, where $x = (x_1, x_2)$, $y = (y_1, y_2)$
b) V = P_3, \langle p(x), q(x) \rangle = p(1)q(1) for p(x), q(x) \in P_3
(Here, p(1) just means set x = 1 in p(x). For example, if p(x) = x^3 - 4x + 1 then p(1) = 1 - 4 + 1 = -2.)
c) V = M_{2,2}, \langle A, B \rangle = det(AB)