Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$ be in $\mathbb{R}^{m \times n}$ such that $\mathrm{rank}(A) = k$ and $A_{11}$ is a $k \times k$ invertible matrix.
Is there a (nonconstant) path $\gamma : (-1,1) \to \mathbb{R}^{m \times n}$ such that $\gamma(t)$ has rank $k$ for each $t$ in $(-1,1)$ and $\gamma(0) = A$?
I'm trying to find the tangent space at $A$ of the submanifold of rank $k$ matrices in $\mathbb{R}^{m \times n}$.