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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For , consider and calculate: . For , consider . Then for some . Letting we see that , so . For the last sentence, observe that and . So equating one subspace with the other, means that their respective direct complements are equal.
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