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Proof of Orthonormal Basis

  1. Oct 1, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove: if an n × n matrix A is orthogonal (column vectors are orthonormal), then the columns form an orthonormal basis for R^n.
    (with respect to the standard Euclidean inner product [= the dot product]).

    2. Relevant equations
    None.


    3. The attempt at a solution

    I know that since the column vectors are orthonormal, all I have to show is that these vectors are also linearly independent and span R^n.

    But I'm having some trouble showing this, so I was thinking about showing it through the basis coordinates:

    u= <u, v1>v1 + <u, v2>v2 +...+ <u, vn>vn

    But I think I have to start with assuming that the vectors v1, v2, ... vn form a basis. So I think that method can't work.
     
  2. jcsd
  3. Oct 1, 2009 #2

    Dick

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    Linear independence is what you want to show. That means c1*v1+c2*v2+...+cn*vn=0 (where vi are your orthonormal vectors and the ci are constants) only has the solution c1=c2=...=cn=0. Can you show that? Once you have that, any set of n linearly independent vectors in R^n is a basis.
     
  4. Oct 1, 2009 #3
    I've been trying to figure out how, but I just can't seem to think of a way...anymore advice?
     
  5. Oct 1, 2009 #4

    Dick

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    You assume c1*v1+c2*v2+...+cn*vn=0. What is vi.(c1*v1+...+cn*vn)? '.'=dot product. Use orthonormality.
     
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