Proof: Preserving Dot Product with Orthogonal Matrices

We can use the properties of the dot product and the definition of an orthogonal matrix to prove this statement.

First, recall that the dot product of two vectors 'u' and 'v' in R^n is defined as:

<u, v> = u^T v

where u^T is the transpose of u.

Now, let A be an nn orthogonal matrix. This means that A^T A = I, where I is the nn identity matrix. Using this property, we can show that:

<Aα, Aβ> = (Aα)^T (Aβ) = α^T A^T A β = α^T I β = α^T β

where we have used the fact that A^T = A^(-1) for an orthogonal matrix.

On the other hand, we have:

<α, β> = α^T β

Therefore, we have shown that <Aα, Aβ> = <α, β> for any n*n orthogonal matrix A.