Dimension of sum of subspaces Let $V$ be a vector space of dimension $29$ over the field $\mathbb{F}$. Suppose that $U$ and $W$ are subspaces of $V$ with $\dim (U) = 24$ and $\dim(W)=15$.
What are the possible values of $\dim (U \cap W)$ and $\dim (U+W)$?  
I know that
$\dim(U+W) = \dim (U) + \dim (W) - \dim (U\cap W) = 39 - \dim(U\cap W)$.
How would I find the range of the dimension of $U+W$ and dimension of $U\cap W$?
 A: The dimension of $U \cap W$ can be no more than the dimension of $U$ nor more than the dimension of $W$. So there is the dimension of $U \cap W$ is at most $15$.
Since both $U$ and $W$ are subspaces of $V$, which has dimension $29$, the dimension of $U+V$ is at most $29$.
Putting these two together, we see that:
\begin{align*}
10 &\leq \dim (U \cap V) \leq 15 \\
24 &\leq \dim (U + V) \leq 29
\end{align*}
I like to think of it as $29$ degrees of freedom available in $V$ and $U+W$ has at least the degrees of freedom occupied by $U$, which is $24$, and cannot surpass $29$. $U \cap W$ can do no more than $W$, which has $15$ degrees of freedom, and, trying to minimize the intersection with $U$, which has already occupied $24$ degrees of freedom, there are only $5$ left in $V$ for $W$, meaning it is forced to share at least $10$ with $U$.
A: The dimension of $U+W$ can be any integer in the interval $[24,29]$: it’s easy to make examples for each possibility.
Once you know the dimension of $U+W$, the dimension of $U\cap W$ follows from Grassmann’s formula, so
$$
\dim(U\cap W)=39-\dim(U+W)
$$
For the examples, consider $\mathbb{F}^{29}$ with its canonical basis $\{e_1,e_2,\dots,e_{29}\}$. Set $U$ the span of $\{e_1,\dots,e_{24}\}$ and $W$ the span of
$$
\{e_{10+k},\dots,e_{24+k}\}
$$
for $0\le k\le 5$.
