Hi Tyler,
I hope you are having a good Spring break.  Indeed, it is not entirely 
clear what now becomes of proposals and presentations.  However, if the 
main objective is to learn some analysis, then I think a large portion 
of the opportunity afforded to do that is in the proposal 
composition/revision process, and I suppose that can pretty much 
proceed as before.  This particular proposal looks to be pretty much 
correct.  The only (relatively minor) modifications I might suggest 
are the following:
(1) Concerning the sets in (ii) and (iii) which are not *necessarily* 
subspaces, you write several times that they are "not subspaces," 
with no qualification.  This is generally pretty much to be avoided 
in careful mathematical exposition.  You can say these are "not 
subspaces in general."  But if you assert that they are simply 
"not subspaces," then the reader expects you to prove they are not 
subspaces in all cases.  Of course, you didn't mean this nor prove this.  
You gave examples and understand very well that there might be 
cases/examples in which these sets might be subspaces.

(2) You seem to be lacking a symbol for relative complement.  
Maybe you tried to use "~" in your word processing software, but 
nothing is showing up.  I would think W_1\W_2 should show up, but 
then perhaps W_1~W_2 might show up as well.  In any case, problems 
of notation are rather nicely dealt with by using tex/latex which is 
pretty standard, easy to implement, and probably worth trying out 
and knowing.
(3) Perhaps some of your language/presentation could be tightened 
up a bit.  For example in part (1) you might compare your presentation 
to something like this:
Let w_1,w_2 \in W_1\cap W_2.  Then w_1,w_2 \in W_j for j = 1,2, and 
therefore, w_1+w_2 in W_j for j = 1,2.  Hence, w_1+w_2 \in W_1\cap W_2, 
and the intersection is closed under addition.  Similarly, if 
\alpha \in F and w\in W_1\cap W_2, then w \in W_j for j = 1,2.  
It follows that alpha w \in W_1\cap W_2 and the intersection is 
closed under scaling. 

I think if you are willing to do so, then I would like to post your 
solution on the course page.  There are various options for doing 
(or not doing) this.  Some of those are:
(a) Post this version (with your name) and my comments above along 
with a revised version incorporating the comments to whatever extent 
you find reasonable.
(b) Post some other anonymous version without your name (with or 
without comments and with our without revision).  (I can probably 
save and make a copy without your name if you want.)
(c) Post nothing. 

Please let me know what you would like to do and what you think 
would be the most instructive thing we can do.