Possible dimensions of $V \cap W$ in $\mathbb{R}^n$

Question

If $V$ and $W$ are two dimensional subspaces of $\mathbb{R}^n$, what are the possible dimensions of $V \cap W$ if;

• $V$ and $W$ have the same dimensions? (for example; $\dim(V)=2, \dim(W)=2, n=3$)
• $V$ and $W$ have different dimensions? (for example; $\dim(V)=4, \dim(W)=5, n=10$)

I am having difficulties to solve this kind of problems. (NOTE: I am not asking you to solve the following two problems, but I am asking about the formulea to be used, then I will try solving them).

I have already saw this, but could not find a general formula.

Answer 1

Since $U\cap V \subset U$ and $U\cap V \subset V$, we have $$\color{blue}{\dim (U\cap V) \le \min \{\dim(U), \dim(V)\}}$$ In case you didn't know, we also have $${\dim(U\cap V) = \dim(U) + \dim(V) - \dim(U+V)}$$ Note that $${n \ge \dim(U+V) \ge \max\{\dim U, \dim V\}}$$ since $V \subset U+V$ and $U \subset U+V$. Combining this with the previous equality, we have $$\color{blue}{\dim(U) + \dim(V) - n \le \dim(U\cap V) \le \dim(U) + \dim(V) - \max\{\dim(U), \dim(V)\}}$$

You should be able to tackle your questions now.