(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

From Calculus on Manifolds by Spivak: 1-7

A Linear Transformation T:R^{n}-> R^{n}is Norm Preserving if |T(x)|=|x| and Inner Product Preserving if <Tx,Ty>=<x,y>.

Prove that T is Norm Preserving iff T is Inner Product Preserving.

2. Relevant equations

T is a Linear Transformation

=> For All x,y [tex]\in[/tex] R^{n}and scalar c

1. T(x+y)=T(x)+T(y)

2. T(cx)=cT(x)

|x|=sqrt((x^{1})^{2}+...+(x^{n})^{2})

x is an n tuple i.e. x=(x^{1},...,x^{n})

<x,y>=[tex]\sum[/tex]x^{i}y^{i}where i=1,...,n

3. The attempt at a solution

I cannot see what to do here at all. I am definately missing something. I don't see how the definitions relate to help me here. It's probably something simple. Any direction or hint would be greatly appreaciated. Thank you.

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# Homework Help: Linear Transformation Norm Preserving

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