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Question about a proof.

  1. Jul 11, 2009 #1
    Suppose T ∈ L(V) and U is a subspace of V. Prove that U is
    invariant under T if and only if U⊥ is invariant under T∗.

    Now for reference, L(V) is the set of transformations that map v (a vector) from V to V.
    T* is the adjoint operator.
    The case where the dimension of U is less than V bugs me. How can U
    be invariant under T* when T* maps from U to V unless mapping to V
    also counts as mapping to U since U is in V itself. Now when
    I say map to V, I mean lets say V is 3 dimensional and U is 1 dimensional.
    Then u=(x3) gets mapped (by T*) to a vector with three nonzero components.
    Does this count as mapping from U to U?
    I'm just a bit confused about this, and any help will be greatly appreciated.
    I guess that I'm confused enough that my post doesn't make much sense,
    so in that case, could someone nudge me in the right direction (ie give a good hint)?
    Thank you!
     
    Last edited: Jul 11, 2009
  2. jcsd
  3. Jul 12, 2009 #2
    Well, ok the thing that I don't understand is how nullT* can even
    exist if T* sends a vector from a space of lower dimension to a space of higher dimension
    unless 0 is the only element in nullT*. So if T* maps a vector from 2d space to 3d space,
    what is within nullT*? Sorry for double posting, but I need to know badly. Thank you!
     
  4. Jul 12, 2009 #3

    Office_Shredder

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    Umm... T* sends a vector from V to V. It's defined to be the linear transformation such that <Tu,v> = <u,T*v>

    The definition of invariance is U is invariant under T if T(u) is in U for all u in U. So we want to show T maps elements of U to U if and only if T* maps elements of U to U. Start with the definition of T* above and the definitino of U
     
  5. Jul 13, 2009 #4
    Let v be a vector in V, let u be a vector in U and let w be a vector in U[tex]\bot[/tex]. Let v=u+w. Now <Tu, u>+<Tu, w>=<u, T*u>+<u, T*w>
    If T*w maps from U[tex]\bot[/tex] to U[tex]\bot[/tex], then <u, T*w>=0 therefore
    <Tu, u>+<Tu, w>=<u, T*u> as a result. Now given that <Tu, u>=<u, T*u> (apply * to both sides gives
    <T*u, u>+<u, T**u>=<T*u, u>+<u, Tu>) this equation holds when w is orthogonal to Tu which forces
    T to map u from U to U as a result of w being orthogonal to Tu and to make <Tu, u>+<Tu, w>=<u, T*u> true.
    Note that we wouldn't have arrived at <Tu, u>+<Tu, w>=<u, T*u> if T* didn't map w from U[tex]\bot[/tex]
    to U[tex]\bot[/tex] and this equation allows <Tu, w> to=0 which allows Tu to map from U to U
    since all elements u in U are orthogonal to all elements w in U[tex]\bot[/tex].
     
    Last edited: Jul 13, 2009
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