Hello. I am having problems with our 4th assignment in linear algebra. It's like I am completely clueless on what to do. I hope you can help me prove these.

(1) LetVbe a vector space over a fieldF. LetWbe a subspace ofV. Prove that dimV= dimW+ dim (V/W) using the definition of a basis.

(2) LetU, V, Wbe finite-dimensional vector spaces over a fieldF.

Let dimU=n; dimV=m; and dimW=p.

Letf:U->U ; g:U->U ; h:U->Vandk:V->Wbe linear transformations.

Show that:

a) rank of (f+g) <= rank off+ rank ofg

b) rank of (kh) <= min{rank ofk, rank ofh}

c) nullity of (kh) <= nullity ofh+ nullity ofk

d) rank off+ rank ofg- n <= rank of (fg) <= min{rank off, rank ofg}

e) nullity of (hg) >= max {nullity ofg, nullity ofh}