Results 1 to 2 of 2

Thread: Linear Algerbra Proof

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    61

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    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 $\displaystyle V,$ 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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Algerbra
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Mar 18th 2009, 04:21 AM
  2. algerbra
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Jan 27th 2009, 08:19 PM
  3. help with algerbra equations please
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Jun 4th 2008, 09:16 PM
  4. Need some help with some linear algerbra
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 19th 2008, 05:09 AM
  5. Algerbra help
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Jan 31st 2008, 06:13 PM

Search Tags


/mathhelpforum @mathhelpforum