[18 pts] Let p(x) = ao+aa+ax and g(x) = bo+bx+b2a. Consider the following two inner products...

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[18 pts] Let p(x) = ao+a₁a+a₂x² and g(x) = bo+bx+b2a². Consider the following two inner products in P₂[x]: (p, q)₁ aobo + a1b₁ + a2b₂ (1) and (p, q)₂p(0)q(0) +p(1)q(1) +p(2)q(2). (2) (a) Let ||-|| be the norm induced by the inner product defined in (1). Show that if p(x) = 1-x+ 2x² and q(x) = -1 + 2x + x2, then ||p|| = ||g||, but ||pg|| #0. (b) Repeat part (a) assuming that ||-|| is the norm induced by the inner product defined in (2) [18 pts] Let p(x) = ao+a₁a+a₂x² and g(x) = bo+bx+b2a². Consider the following two inner products in P₂[x]: (p, q)₁ aobo + a1b₁ + a2b₂ (1) and (p, q)₂p(0)q(0) +p(1)q(1) +p(2)q(2). (2) (a) Let ||-|| be the norm induced by the inner product defined in (1). Show that if p(x) = 1-x+ 2x² and q(x) = -1 + 2x + x2, then ||p|| = ||g||, but ||pg|| #0. (b) Repeat part (a) assuming that ||-|| is the norm induced by the inner product defined in (2)