Please excuse my previous post as I accidentally pressed the submit button instead of review button. English is not my first language but I'm doing my best. I hope that the following helps, I like to avoid using any vectors whatsoever in proofs like this.

Ok. first (i)

(ii). Given (i) then you know that

. We also know that

. It is known that the dimesion of a subspace is lesser than or equal to the dimesion of the larger space. Hence we have shown that

.

Now

. Given (ii) we automatically know that

, and we know that

. Next, we know that since

and

are both subspaces of

then their sum (that is

) is also a subspace of

, hence the following holds:

.

Finally, since we have the equation

, which gives us that

since

. Effecetively we have shown that

also holds. Then we must have

. This gives us that

and, since

, therefore

which is equivalent to (i).