1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
    |T(h)|=sqrt((a11h1+...+a1nhn)2+...+(am1h1+...+amnhn)2)
    where aij's are the elements in the matrix representation of T, A.

    |h|=sqrt((h1)2+...+(hn)2)

    :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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Linear Transformation and Magnitudes
  1. Linear Transformation (Replies: 1)

  2. Linear transformations (Replies: 4)

  3. Linear transformation (Replies: 4)

  4. Linear transformation (Replies: 18)

Loading...