Given are two subspaces $U$ and $W$ of subspace $V.$
\begin{align}
\dim(V) & = 7 \\ \dim(U) & = 4 \\ \dim(W) & = 5
\end{align}
what is the dimension of $U \cap W$ ?
I know that the Dimension of subspaces cannot be bigger then the one containing them.
I also know that $\dim(U+W) = \dim U + \dim W – \dim(U \cap W)$
so I'm assuming that the intersection will be small or equal to $4$ on one end since thats the size of the smaller of $U$ and $W$ but I'm not sure about the the other end.
Best Answer
You're very much on the right track. If you put together the following two facts: $$\dim(U+W)=\dim(U)+\dim(W)-\dim(U\cap W)$$ and $$\dim(U+W)\le\dim(V)=7,$$ you'll get an inequality that will give you the least possible value for $\dim(U\cap W)$. So the answer will be that $$\cdots\le\dim(U\cap W)\le4,$$ where you just need to find that lower number.