Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.
(i) Show that im(S+T) is a subset of im(S) + im(T)
(ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.
(i) Show that im(S+T) is a subset of im(S) + im(T)
(ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
I'll help with i) and the first part of ii) and leave the rest to you. For i) merely note that if $\displaystyle v\in\text{im}\left(S+T\right)$ then $\displaystyle v=(S+T)(w)$ for some $\displaystyle w\in V$. But, $\displaystyle (S+T)(w)=S(w)+T(w)\in \text{im}(S)+\text{im}(T)$...conclude.
For the first part of ii) the fact that $\displaystyle \text{rk}(ST)\leqslant \text{rk}(S)$ is clear since $\displaystyle \text{im}(ST)=(ST)(V)=S(T(V))\subseteq S(V)=\text{im}(S)$. Now, to see that $\displaystyle \text{rk}(ST)\leqslant \text{rk}(T)$ note that in general the image of a linear homomorphism has lesser dimension than the domain etc.