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

1. Problem 1-8(b)

If there is a basis x_{1},..., x_{n}of R^{n}and numbers a_{1},..., a_{n}such that Tx_{i}= a_{i}x_{i}(where T is a linear operator), prove that T is angle-preserving if and only if all the |a_{i}| are equal.

2. Relevant equations

First a couple of definitions...

Definition (i): If x,y are nonzero vectors in R^{n}, the angle between them is given by

angle(x,y) = arccos {<x,y>/ ( |x | |y | )}

where < , > denotes the standard inner product and | | the standard norm.

Definition (ii): A linear transformation T is angle-preserving if T is 1-1 and for x,y != 0 we have that angle(Tx, Ty) = angle(x,y).

3. The attempt at a solution

Problem is, I think I see a simple counterexample to the sufficiency part of the proof. Let x_{1}, x_{2}be a basis for R^{2}where

x_{1}= e_{1}(i.e. standard unit basis vector) and

x_{2}= e_{1}+ e_{2}

Then suppose Tx_{1}= x_{1}and Tx_{2}= -x_{2}. Now consider the angle between x_{1}and x_{1}+x_{2}:

angle(x_{1}, x_{1}+x_{2}) = angle(e_{1}, 2e_{1}+e_{2}) ~ 25 degrees, but

angle(Tx_{1},T(x_{1}+x_{2})) = angle(x_{1}, x_{1}-x_{2}) = angle(e_{1}, -e_{2}) = 90 degrees.

So even though the absolute values of a_{1}= 1 and a_{2}= -1 are equal, it seems T does not preserve angles. What am I misunderstanding about this problem? I am assuming the author means angle-preserving with respect to the standard inner product in which <e_{1}, e_{2}> = 0.

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# Homework Help: Spivak's calculus on manifolds chapter 1

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