If $T:V\to V$ be linear trannsformation on a vector space $V$ with rank$(T)\le$ rank$(T^3)$. Then, what can be said about the null space and range of $T$. Typically, do they have empty intersection, or, is one the subset of another, or does there exist a subspace $W$ of $V$which equals their intersection.
I know that range of $T^3$ is a subspace of $T$ and null space of $T$ is a subspace of $T^3$. But, am unable to proceed further. Any hints. Thanks beforehand.
Best Answer
well first of all, lets consider $T$ as a linear map (this makes stuff in my opinion often a lot clearer). We know that $rank(T)=\dim \textrm{im}(T)$, But we know that $\text{im}(T^3)=\textrm{im}(T\mid_{\textrm{im}(T^2)})$, in particular $\textrm{im}T^3\subset \textrm{im} T$ but that just means in dimensions that $\dim \textrm{im}(T) \ge \dim \textrm{im}(T^3) $ in particular we get with your inequality $$\dim \textrm{im}(T) = \dim \textrm{im}(T^3)$$ But this just means that $T\mid_{\textrm{im}(T)}$ is an isomorphism. So $T\mid_{\textrm{im}(T)}$ has kernel zero. But this immediately gives $$0=\ker(T\mid_{\textrm{im}(T)}) = \ker(T)\cap \textrm{im}{T}$$