# Proof of Orthonormal Basis

1. Oct 1, 2009

### abbasb

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. Oct 1, 2009

### Dick

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. Oct 1, 2009

### abbasb

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

4. Oct 1, 2009

### Dick

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