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Homework Help: Vector Spaces and Correspondence

  1. Oct 8, 2012 #1
    1. The problem statement, all variables and given/known data
    This question came out of a section on Correspondence and Isomorphism Theorems

    Let [itex]V[/itex] be a vector space and [itex]U \neq V, \left\{ \vec{0} \right\} [/itex] be a subspace of [itex]V[/itex]. Assume [itex]T \in L(V,V)[/itex] satisfies the following:
    a) [itex]T(\vec{u} ) = \vec{u}[/itex] for all [itex]\vec{u} \in U[/itex]
    b) [itex]T(\vec{v} + U) = \vec{v} + U[/itex] for all [itex]\vec{v} \in V[/itex]
    Set [itex]S=T-I_{V}[/itex]. Prove that [itex]S^{2}=\vec{0}_{V \rightarrow V}[/itex]

    2. Relevant equations
    [itex]I_{V}[/itex] is the identity map
    [itex]L(V,V)[/itex] is the map of all linear operators on V

    3. The attempt at a solution
    I have trouble understanding the question.
    Since [itex]T \in L(V,V)[/itex] then how is [itex]T(\vec{v} + U) = \vec{v} + U[/itex] for all [itex]\vec{v} \in V[/itex]?
    Wouldn't that mean [itex]T:V/U \rightarrow V/U[/itex]?
    I don't understand, what is [itex]T(\vec{v})[/itex] equal to?
    Does [itex]T(\vec{v})=\vec{v}[/itex] or [itex]T(\vec{v}) = [\vec{v}]_W[/itex] or something else?

    I'm sorry if this is a silly question.
  2. jcsd
  3. Oct 8, 2012 #2
    With [itex]T(v+U)=v+U[/itex], they mean that the set v+U is mapped to the set v+U.

    [tex]T(\{v+u~\vert~u\in U\})=\{v+u~\vert~u\in U\}[/tex]

    Or in another way: for each [itex]v\in V[/itex] and [itex]u\in U[/itex], there exists [itex]u^\prime\in U[/itex] such that [itex]T(v+u)=v+u^\prime[/itex].
  4. Oct 8, 2012 #3
    This is where I confuse myself. T is a linear transformation from V to V. If it maps the set v+U to the set v+U, then wouldn't that be mapping cosets of V mod U to cosets of V mod U instead of mapping V to V?
  5. Oct 9, 2012 #4
    I think I get it now. The function maps the coset of V mod U to the same coset of V mod U by mapping each individual element to another element in that coset.

    I'm sorry if it seemed I brushed over your post and didn't completely read it. I did. I just didn't understand it. I'm having trouble with my Linear Algebra 2 class, and I'm glad that you responded. Thank you a lot!

    So then I get
    Let [itex]\vec{v} \in V[/itex] be arbitrary
    [itex]S^2(\vec{v} + \vec{u}), \vec{u} \in U[/itex]
    [itex]=S(T(\vec{v} + \vec{u}) - I(\vec{v} + \vec{u}))[/itex]
    [itex]=S(\vec{v} + \vec{u}` - ( \vec{v} + \vec{u}))[/itex] where [itex]\vec{u}` \in U[/itex]
    [itex]=S(\vec{u}` - \vec{u})[/itex]
    [itex]=S(\vec{y})[/itex] where [itex]\vec{y} = \vec{u}`- \vec{u} \in U[/itex]
    [itex]=T(\vec{y}) - I(\vec{y})[/itex]
    [itex]=\vec{y} - \vec{y}=\vec{0}[/itex]
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