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Homework Statement
Suppose that A is an n x n matrix such that A(Transpose)A=I. Let x be any vector in R^n. Show that llAxll=llxll; that is, multiplication of x by A produces a vector Ax having the same length as x.
Homework Equations
Sqrt(x(transpose)x)=llxll
The Attempt at a Solution
So I started setting up the equation wth an n x n matrix...
l a b l =A
l c d l
l a c l= A (transpose)
l b d l
A(transpose)A= I=
l a^(2) +c^(2) ab+cd l
l ba+dc b^(2)+d^(2)l
then I let the vector x=
l x1 l
l x2 l
then from there I was going to use the distance formula on Ax
but I wasn't sure if doing the dot product of Ax is the same thing as matrix multiplication of Ax, and my second problem is that I am not sure how to take the square root of a 2x2 matrix if they are the same thing.
If someone can answer those to questions for me I think I can finish the problem.
Thank you.