Dimension of vector space in extreme cases. Let V be a vector space of dimension 29 over a field $\mathcal{F}$.
Suppose that U and W are subspaces of V with dim(U) = 24 and dim(W) = 15 
1) What are the possible values of dim(U+W)?
My reasoning on this is since we know $\dim(U+W) = \dim(U)+\dim(W) - \dim(U\cap W)$
the maximum of dim(U+W) would be when $\dim(U\cap W) = 0$ 
ie $24+15=39$ and the minimum would be when $\dim(U\cap W)$ attains
a 
maximum value ie $15$, so the range of values would be 
$24-39$.
2) Describe U+W in the two extreme cases (that is when $\dim(U\cap W)$
takes
the least and greatest values possible?.) 
This I am not so sure about, if $\dim(U\cap W)=\{0\}$, I know we call
the $U+W$ the direct sum 
denoted $U\oplus W$, I think it means if $\dim(U\cap W)=\{0\}$
the two bases are linearly independent?, 
but I am not sure what the other extreme case describes?. 
3) Write down explicit subspaces of $U,W$ of $V$ = $\mathbb{R}^{29}$
with 
$\dim(U)= 24$ $\dim(W)=15$ and $\dim(U\cap W)=12$. 
I'm unsure how to make a start on this question?. 
 A: 1) Since $24+15>29$ you can't have non-intersecting $U$ and $W$, the best you can do is minimize the dimension of their intersection to $24+15-29=10$. The range is 24-29.
2) For the first extreme take $10$ first standard basis vectors to be a basis of $U\cap W$. Of remaining 19 add 14 to those 10 to span $U$, and add the remaining 5 to span $W$. 
For the second extreme you should have $U\cap W=W$, in other words, $W$ has to be a subspace of $U$. Chose first 15 basis vectors to span $W$, and add the next $9$ to span $U$.
3) The same idea as in 2), span $U\cap W$ with the first $12$ basis vectors, this leaves $29-12=17$ still available. Of them $12$ go into $U$ for the total of $24$, and $3$ into $W$ for the total of $15$. 
A: For 1, if $U,W$ are subspaces of $V$, their sum is also a subspace of $V$, so its dimensionality cannot exceed that of $V$.  
For 2, note that the dimension of $U$ is the same as that of $(U+W)$ and all the vectors in $U$ are clearly in $(U+W)$ because $0$ is in $W$  
For 3, can you write down $24$ independent basis vectors for $U$ to get started?
