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Homework Help: Linear Transformation and Magnitudes

  1. Feb 28, 2009 #1
    1. The problem statement, all variables and given/known data
    From Calculus on Manifolds by Spivak: 1-10
    If T:Rm -> Rn is a Linear Transformation show that there is a number M such that |T(h)| [tex]\leq[/tex] M|h| for h[tex]\in[/tex]Rm

    2. Relevant equations
    T is a Linear Transformation
    => For All x,y [tex]\in[/tex] Rn and scalar c
    1. T(x+y)=T(x)+T(y)
    2. T(cx)=cT(x)

    3. The attempt at a solution
    Well I didn't get very far but I do know this. The matrix of T with respect to the standard basis is A. Ah=T(h)
    So we can write what |T(h)| and |h| look like.
    where aij's are the elements in the matrix representation of T, A.


    :confused: Suggestions, hints and clues are all appreciated! :smile:
  2. jcsd
  3. Mar 1, 2009 #2
    Try to show that the function [tex]f:\mathbb{R}^m\to\mathbb{R}[/tex], [tex]f(x)=|T(x)|[/tex] has a maximum when restricted to the unit sphere [tex]S^{m-1}[/tex] in [tex]\mathbb{R}^m[/tex]. Then use the fact that every nonzero vector can be written in the form [tex]\lambda v[/tex], where [tex]\lambda[/tex] is a scalar and [tex]v[/tex] is a unit vector.
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