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!

Norm of a linear transformation

  1. Jan 27, 2010 #1
    1. The problem statement, all variables and given/known data
    ||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1}



    3. The attempt at a solution

    {max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1}

    does that look right? I need to show equality...
     
  2. jcsd
  3. Jan 27, 2010 #2

    radou

    User Avatar
    Homework Helper

    I assume you are speaking of a bounded linear transformation?

    If T is bounded, then there exists some constant C so that ||Tx||<=C||x|| for all x from the domain of T, and it follows almost directly from this definition.
     
  4. Jan 27, 2010 #3
    You're on the right track. What is the relation between the vector [tex]Tx[/tex] and the vector [tex]T\left(\frac{x}{\|x\|}\right)[/tex] ?
     
  5. Jan 27, 2010 #4
    I don't know that T is bounded...T is on R^n


    Tx >= T(x/||x||)
     
  6. Jan 27, 2010 #5
    [tex]Tx[/tex] and [tex]T\left(\frac{x}{\|x\|}\right)[/tex] are vectors in the range space of [tex]T[/tex]; they do not possess an order relation. The relationship between these two vectors is simpler, and follows directly from the definition of a linear transformation.
     
  7. Jan 27, 2010 #6

    (1/||x|| ) Tx = T(x/||x||)
     
  8. Jan 27, 2010 #7

    Landau

    User Avatar
    Science Advisor

    Exactly, by linearity! So, what can you conclude? Try to think a bit for yourself.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Norm of a linear transformation
Loading...