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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

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

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^{T}A = I, as well. thus:

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

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

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

thank You