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Math Help - n(t) and r(t) problem LIN TRANSFORMATIONS

  1. #1
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    n(t) and r(t) problem LIN TRANSFORMATIONS

    7. Suppose that T : V ! V is a linear transformation. Show that N(T) is a subspace of
    N(T^3), and R(T^3) is a subspace of R(T). Show also that N(T) = N(T^3) if and only if
    R(T) = R(T^3)

    Can any1 solve this?
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  2. #2
    Super Member Rebesques's Avatar
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    For N(T)\subset N(T^3), consider x\in N(T) and calculate: T(x)=0\Rightarrow T(T(x))=T^2(x)=0\Rightarrow T^3(x)=0\Rightarrow x\in N(T^3).

    For R(T^3)\subset R(T), consider x\in R(T^3). Then x=T^3(y) for some y\in V. Letting z=T^2(y) we see that x=T(z), so x\in R(T).

    For the last sentence, observe that V=N(T)\oplus R(T) and V= N(T^3)\oplus R(T^3). So equating one subspace with the other, means that their respective direct complements are equal.
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