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Math Help - Linear Algebra please help.

  1. #1
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    Linear Algebra please help.

    Let V, W, and Z be vector spaces, and let T: V-->V. Prove that T[T(v)]= 0 if and only if R(T) is a subset of N(T).
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  2. #2
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    Quote Originally Posted by JCIR View Post
    Let V, W, and Z be vector spaces, and let T: V-->V. Prove that T[T(v)]= 0 if and only if R(T) is a subset of N(T).
    Step 1:

    CLAIM: R(T) \subset N(T) \Rightarrow \forall v \in V, T[T(v)]= 0

    Proof:
    \forall v \in V, T(v) \in R(T) \subset N(T) \Rightarrow T(v) \in N(T) \Rightarrow T(T(v)) = 0
    --------------------------------------------------------------------------------------------------------

    Step 2:
    CLAIM: \forall v \in V, T[T(v)]= 0 \Rightarrow R(T) \subset N(T)

    Proof:
    \forall x \in R(T),\exists v \in V, x = T(v), \text{ but } T(T(v)) = T(x) = 0  \Rightarrow \forall x \in R(T), x \in N(T) \Rightarrow R(T) \subset N(T)
    Last edited by Isomorphism; July 14th 2008 at 12:54 AM.
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