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Spivak's calculus on manifolds chapter 1

  1. Jul 4, 2008 #1
    1. The problem statement, all variables and given/known data
    1. Problem 1-8(b)

    If there is a basis x1,..., xn of Rn and numbers a1,..., an such that Txi = aixi (where T is a linear operator), prove that T is angle-preserving if and only if all the |ai| are equal.



    2. Relevant equations
    First a couple of definitions...

    Definition (i): If x,y are nonzero vectors in Rn, 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 x1, x2 be a basis for R2 where

    x1 = e1 (i.e. standard unit basis vector) and
    x2 = e1 + e2

    Then suppose Tx1 = x1 and Tx2 = -x2. Now consider the angle between x1 and x1+x2:

    angle(x1, x1+x2) = angle(e1, 2e1+e2) ~ 25 degrees, but

    angle(Tx1,T(x1+x2)) = angle(x1, x1-x2) = angle(e1, -e2) = 90 degrees.

    So even though the absolute values of a1 = 1 and a2 = -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 <e1, e2> = 0.
     
    Last edited: Jul 4, 2008
  2. jcsd
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