Show that the set of rank two matrices in $ M_{2X3}($$R$$)$ is open.
Can someone explain, what kind of topology we are talking about here. I feel like $ M_{2X3}($$R$$)$ should be homeomorphic to $R^6$, but then can we claim that every basis of $ M_{2X3}($$R$$)$ will be homeomorphic to some open set of $R^6$
The topology here is indeed the standard topology on $\mathbb{R}^6$. To show that said set is open, I will give you a hint: a matrix $B \in M_{2 \times 3}(\mathbb{R})$ has rank $2$ if and only if a $(2 \times 2)$-minor is non-zero. A minor is a determinant of a submatrix. What can you say about the determinant as a function from the matrices to the reals?