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 then for some . But, ...conclude.
For the first part of ii) the fact that is clear since . Now, to see that note that in general the image of a linear homomorphism has lesser dimension than the domain etc.