Supose α βeR show that Aα Aβ= α β halds for any nn orthogond matrix A

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