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Homework Help: T-Invariant Subspaces

  1. Jun 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that W is a T-invariant subspace of T for:
    W = E[tex]_{\lambda}[/tex]


    2. Relevant equations


    3. The attempt at a solution

    Ok, so I know that I need to show that T maps every element in E[tex]_{\lambda}[/tex] to .

    E[tex]_{\lambda}[/tex] = N(T-[tex]\lambda[/tex]I)

    so T must map every eigenvector related to [tex]\lambda[/tex] to another eigenvector in E[tex]_{\lambda}[/tex]

    T(x) maps to zero vector, when x is an eigenvector associated with [tex]\lambda[/tex] which is in the eigenspace of [tex]\lambda[/tex], correct?
     
    Last edited: Jun 4, 2010
  2. jcsd
  3. Jun 4, 2010 #2

    Office_Shredder

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    Your post is just a huge blank space where you messed up the tex for making a lambda. Can you edit it?
     
  4. Jun 4, 2010 #3
    fixed
     
  5. Jun 4, 2010 #4

    Office_Shredder

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    Ok I get it now. [tex]E^{\lambda}[/tex] is the set of all eigenvectors with eigenvalue [tex]\lambda[/tex]. Let's say v is an eigenvector, and [tex] Tv=\lambda v[/tex]. What is [tex]T(\lambda v)[/tex] and how does this help you answer the question?
     
  6. Jun 4, 2010 #5
    T([tex]\lambda[/tex]v) = [tex]\lambda[/tex]2v

    and this is just a multiple of T, so T(v) = [tex]\lambda[/tex]v maps to the eigenspace?
     
  7. Jun 4, 2010 #6
    and another question...

    Show that W is a T-invariant subspace when

    W = N(T)

    So N(T) : { x E W: T(x) = 0} , since W is a subspace it contains the zero vector, thus any vector in N(T) will map to zero which is in W?
     
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