Show that the following does not define an inner product on $\displaystyle R^3$:

$\displaystyle

<(x,y,z),(a,b,c)> = xa + 2xb + 2ay + yb + 4zc$

Define an inner product on $\displaystyle P_2$ by

$\displaystyle <p,q> = \int^1_{-1} p(x)q(x) dx$

Let $\displaystyle U \subseteq P_2$ be the subspace spanned by {$\displaystyle x,x^2$}.

(i) Find an orthornormal basis for U.

(ii) Find the polynomial in U that is as close as possible to 1. (for the norm corresponding to the above inner product)

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And a little vector space question,

Determine whether the polynomial

$\displaystyle p(x) = x^2 + x + 2$

belongs to span {$\displaystyle {p_1(x), p_2(x),p_3(x)}$} where

$\displaystyle p_1(x) = 2x^2 + x + 2, p_2(x) = x^2 - 2x, p_3(x) = 5x^2 - 5x + 2$

Thanks for your help guys. (: