Let $M_r\subset\mathbb{R}^{n\times n}$ be the manifold of rank $r$ matrices. Let $\mathbb{V}\subset\mathbb{R}^{n\times n}$ be a subspace of dimension $m$. Let $X\in M_r\cap\mathbb{V}$ be a given matrix and we denote the tangent space at $X$ with respect to the manifold by $\mathbb{T} = \mathbb{T}_{X}M_r$.
So my question is: is $M_r\cap\mathbb{V}$ still a manifold?
And if the answer was yes, what can we say about the relation between the dimension of $M_r\cap\mathbb{V}$ and dimension of $\mathbb{V}\cap\mathbb{T}$?