# Rank in Linear Maps

• Feb 8th 2010, 09:31 AM
pseudonym
Rank in Linear Maps

Let S:U -> V, T:V->W be linear maps, where U, V, W are vector spaces over the same field K.
Prove (i) Rank(TS) <= Rank (S)
(ii) If V=W and T is non singular, then Rank(TS) = Rank(S)

Any help would be greatly appreciated!
Thank you
• Feb 8th 2010, 01:46 PM
Roam
Quote:

Originally Posted by pseudonym
We have $dim(T(S(U))) \leq dim S(U)$. Hence $rank(TS) = dim(TS(U))= dim(T(S(U))) \leq dim S(U) = rank S$.