1. ## Linear Algerbra Proof

Need help on this proof!! thanx

Let a, b, and c be distinct real numbers. Show that

<p(x), q(x)> = p(a)q(a) + p(b)q(b) + p(c)q(c)

defines an inner product on P^2.

And the hint is you will need the fact that a polynomial or degree n has at most n zeros.

2. Originally Posted by aeubz
Need help on this proof!! thanx

Let a, b, and c be distinct real numbers. Show that

<p(x), q(x)> = p(a)q(a) + p(b)q(b) + p(c)q(c)

defines an inner product on P^2.

And the hint is you will need the fact that a polynomial or degree n has at most n zeros.
so i guess $\displaystyle V,$ your vector space, is over $\displaystyle \mathbb{R}$ and it consists of all polynomials of degree at most 2. it's obvious that $\displaystyle <,>$ is bilinear. also for any

$\displaystyle p(x) \in V: \ <p(x),p(x)>=(p(a))^2 + (p(b))^2 + (p(c))^2 \geq 0.$ the only thing left is to prove that $\displaystyle <p(x),p(x)>=0$ if and only if $\displaystyle p(x) \equiv 0.$

now $\displaystyle <p(x),p(x)>=0$ if and only if $\displaystyle (p(a))^2 + (p(b))^2 + (p(c))^2=0$ if and only if $\displaystyle p(a)=p(b)=p(c)=0.$ but this means that $\displaystyle p(x),$ which is

a polynomial of degree at most 2, has three distinct zeros! this is possible only if $\displaystyle p(x)$ is the zero polynomial. this completes the proof.