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Math Help - Direct sum of a range and T-invariant subspace problem

  1. #1
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    Direct sum of a range and T-invariant subspace problem

    Hi!
    I've been working for this problem for quite a while, but am seriously stuck now, so I would appreciate any help!
    Here's the problem:
    Suppose  V=R(T) \oplus W and W is T-invariant. Prove that  W \subseteq N(T)

    where R(T) is the range of linear transformation T:V \rightarrow V and N(T) is a null space of T, and W is a subspace of V.
    Here's what I got:
    I need to show that for all w \in W, w \in N(T) and w is in N(T) if and only if T(w)={0}, so what i need to show is for all w \in W, T(w) = 0 .
    Since V is a direct sum of R(T) and W, then  V=R(T)+W and  R(T) \cap W = \{0\} . Since V=R(T)+W, i know that for all  v \in V , there exists  y \in R(T) and there exists  w \in W such that v=y+w. Since there exists y in R(T), then there exists an  x \in V such that T(x)=y. And from W being T-invariant, i know  T(w) \in W .
    Ok, knowing it is good, but i cannot relate that to N(T). What do I do next??

    Thank you for any help!!
    Last edited by vanishingpoint; March 30th 2010 at 04:49 PM.
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  2. #2
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    If we take w \in W, then from W being T-invariant, we get that T(w) = w' for some w' \in W; that is, T(w) \in W.

    But we also know that T(w) \in Range(T) by definition, therefore we get that T(w) \in W \cap Range(T) but this is possible only if T(w) = 0 \Rightarrow w \in Null(T) \Rightarrow W \subset Null(T)
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  3. #3
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    Oh, thank you so much! it is so easy now that you explained it! makes perfect sense. Thanks a lot!
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