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

1. ## 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?

2. 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.