Linear subspaces dimensions (2) I have the following question:

$U$ and $V$ are two linear subspaces of $\mathbb{R}^7$, with $\dim(U) = \dim(V) = 5$.
What is the minimum dimension of $U ∩ V$?


Using the Grassmann Formula I have:
$\dim(U ∩ V) + \dim(V + U) =  \dim (U) + \dim(V)$
$\dim(U ∩ V) + \dim(V + U) = 5 + 5 = 10$
$\dim(U ∩ V) = 10 - \dim(V + U)$
So to answer I have to find the maximum dimension of $\dim(V + U)$.
Because $V + U ⊆ ℝ^7$, the max dimension is $7$. So:
$\dim(U ∩ V) = 10 - \dim(V + U) = 10 - 7 = 3$
Is this right? If yes, are there other ways to solve it?

You're correct. By Grassmann's formula,
$$
\dim(U\cap V)=\dim U+\dim V-\dim(U+V)
$$
so it is minimal when $\dim(U+V)$ is maximal. Since it is possible that $\dim(U+V)=7$ (an example would be needed), but not more, the minimum value for $\dim(U\cap V)$ is $5+5-7=3$.
