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Math Help - Linear Algerbra Proof

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by aeubz View Post
    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 V, your vector space, is over \mathbb{R} and it consists of all polynomials of degree at most 2. it's obvious that <,> is bilinear. also for any

    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 <p(x),p(x)>=0 if and only if p(x) \equiv 0.

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

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