Inner Product - Functions.

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June 17th 2008, 05:39 PM
pearlyc

Inner Product - Functions.

Show that the following does not define an inner product on $R^3$ :
$<br /> <(x,y,z),(a,b,c)> = xa + 2xb + 2ay + yb + 4zc$

Define an inner product on $P_2$ by

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

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

(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

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

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

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

Thanks for your help guys. (:
June 17th 2008, 06:16 PM
o_O

Quote:

Originally Posted by pearlyc

Determine whether the polynomial

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

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

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

Your set of polynomials (vectors) span p(x) if p(x) can be written as a linear combination of the vectors $p_1(x), p_2(x), p_3(x)$ , i.e.
$p(x) = ap_1(x) + bp_2(x) + cp_3(x)$
$x^{2} + x + 2 = (2a + b + 5c)x^{2} + (a - 2b - 5c)x + 2(a+c)$

So you have to determine whether or not the system of equations (constructed by equating the coefficients of each term) is consistent or not:
$\begin{array}{ccccccc} 2a & + & b & + & 5c & = & 1 \\ a & - & 2b & - & 5c & = & 1 \\ a & & & + & c & = & 1 \end{array}$

Note that you don't actually have to find the solutions. The determinant will determine whether or not your system is consistent (implying your set of vectors span p(x)).
June 17th 2008, 10:27 PM
Reckoner

Quote:

Originally Posted by pearlyc

Show that the following does not define an inner product on $R^3$ :
$<br /> <(x,y,z),(a,b,c)> = xa + 2xb + 2ay + yb + 4zc$

All of the axioms should hold except this one: $\langle\textbf{v},\;\textbf{v}\rangle\geq 0$ . Think about it. It should be fairly easy to find a counterexample.