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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.