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?
For $\displaystyle N(T)\subset N(T^3)$, consider $\displaystyle x\in N(T)$ and calculate: $\displaystyle T(x)=0\Rightarrow T(T(x))=T^2(x)=0\Rightarrow T^3(x)=0\Rightarrow x\in N(T^3)$.
For $\displaystyle R(T^3)\subset R(T)$, consider $\displaystyle x\in R(T^3)$. Then $\displaystyle x=T^3(y)$ for some $\displaystyle y\in V$. Letting $\displaystyle z=T^2(y)$ we see that $\displaystyle x=T(z)$, so $\displaystyle x\in R(T)$.
For the last sentence, observe that $\displaystyle V=N(T)\oplus R(T)$ and $\displaystyle V= N(T^3)\oplus R(T^3)$. So equating one subspace with the other, means that their respective direct complements are equal.