We all know of real projective space . It is in fact a special space of the Grassmannian manifold, which denoted , is the set of -dimensional subspaces of . Such can be represented via the ranges of the matrices of rank . On application of that operator we can apply any and the range will stay the same. Partitioning by range, we introduce the equivalence relation by if there exists such that . This Grassmannian can be identified with .
Now we find the charts of it. There must be a minor with nonzero determinant. We can assume without loss of generality (as swapping columns changes not the range) that the first minor made of the first columns is one of such, for the convenience of writing , where the is . We get
Thus the degrees of freedom are given by the matrix on the right, so . If that submatrix is not the same between two full matrices reduced via inverting by minor, they cannot be the same as an application of any non identity element in would alter the identity matrix on the left.
I’ll leave it to the reader to run this on the real projective case, where .