inner product by A is the same as the standard inner product on R^n

The Question is as follows

Attachment 25987

Please some on help I really don't know how to do this one.

Thank you in advanced

what you "probably" mean by the inner product induced by A is:

<u,v> = (Au)^T(Av).

if AA^T = I, then A^T = A^-1, so A^TA = I, as well. thus:

<u,v> = (Au)^T(Av) = (u^TA^T)(Av) = u^T(A^TA)v = u^Tv = u.v

You have really done me a great favor by helping me with my two questions
thank You