1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Lineaer algebra - orthogonality

  1. Feb 8, 2013 #1
    1. The problem statement, all variables and given/known data
    Let<x,y> be an inner product on a vector space V, and let e1, e2,...,en be an orthonormal basis for V.
    Prove: <x,y> = <x,e1><y,e1>+...+<x,en><y,en>.

    2. Relevant equations

    <x,x> = abs(x)
    a<x,y> = <ax,y> = <x,ay>

    3. The attempt at a solution

    = x1y1+...+xnyn
    = x1<e1,e1>y1<e1,e1>+....+xn<en,en>yn<en,en>
    * <e,e>=1

    And i cannot find a way to take the e2 out of the equation.
    Any help is appreciated
  2. jcsd
  3. Feb 8, 2013 #2


    User Avatar
    Homework Helper

    How is an orthonormal basis defined?

    A subset {v_1,...,v_k} of a vector space V, with the inner product <,>, is called orthonormal if <v_i,v_j>=0 when i≠j. That is, the vectors are mutually perpendicular. Moreover, they are all required to have length one: <v_i,v_i>=1.

  4. Feb 8, 2013 #3
    Orthonormal basis
    A set of vectors for an inner product space whose are linearly independence and span V and orthonormal.

    orthonormal: orthogonal (<x,y>=0) and each vector has length one.
  5. Feb 8, 2013 #4


    User Avatar
    Science Advisor

    Under "relevant equations" you have "<x, x>= abs(x)". There is NO "x2" so what, exactly do you mean by "en2"? <en, en>?
    You also say "orthonormal: orthogonal (<x,y>=0) and each vector has length one."
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook