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One to one linear operator

  1. Oct 29, 2011 #1
    1. The problem statement, all variables and given/known data

    Let T be a linear operator on a finite dimensional vector space V. Suppose ||T(x)|| = ||x|| for all x in V, prove that T is one to one.

    2. Relevant equations

    ||T(x)||^2 = <T(x),T(x)>
    ||x||^2 = <x,x>

    3. The attempt at a solution

    Suppose T(x) = T(y) x, y in V
    Then ||T(x)|| = ||T(y)||
    and so <T(x),T(x)> = <T(y),T(y)>
    By assumption, we have <x,x> = <y,y>
    But then i cant proceed on to show that x=y
     
  2. jcsd
  3. Oct 29, 2011 #2

    Mark44

    Staff: Mentor

    One form of the definition for one-to-one is this:
    For all x and y in V, T(x) = T(y) ==> x = y.
    You can do a proof by contradiction by assuming that for some x and y in V, T(x) = T(y) and x [itex]\neq[/itex] y.

    You need to use the fact that T is a linear operator.
     
  4. Oct 29, 2011 #3
    If you've seen the right theorem, proving that [itex] \ker(T) = \{0\} [/itex] might be a slightly easier way to do this.
     
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