Proof of Orthonormal Basis

  • Thread starter abbasb
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  • #1
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Homework Statement



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]).

Homework Equations


None.


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.
 

Answers and Replies

  • #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.
 
  • #3
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I've been trying to figure out how, but I just can't seem to think of a way...anymore advice?
 
  • #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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