Affine subspaces of matrices with rank in a range

For every m,n ∈ N and every field K, let M(m × n, K) be the vector space of the (m × n)-matrices over K and let $A(n,K)$ be the vector space of the antisymmetric (n × n)-matrices over K. Let r, s ∈ with s ≤ r. Define AK(m × n;s,r)= \ arrayll S \;| & \; S \; affine subspace ofM(m × n, K)such that & s =min (A)| \; A ∈ S \, \; r =max (A)| \;, A ∈ S \ array \ AantisymK(n;s,r)= \ arrayll S \;| & \; S \; affine subspace ofA(n,K)such that & s =min (A)| \; A ∈ S \, \; r =max (A)| \; A ∈ S \ array \, AechelK(m × n;s,r)= \ arrayll S \;| & \; S \; affine subspace ofM( m × n,K)such that \\ & A \; is in row echelon form ∀ A ∈ S & s =min (A)| \; A ∈ S \, \; r =max (A)| \; A ∈ S \ array \. In this paper we prove the following formulas: if |K| ≥ r+2 and the characteristic of K is different from $2$, then max \(S) \; |\; S ∈ AK(m × n; s,r)\ = r max ,n\ - s+12 and max \(S) \; |\; S ∈ AantisymK( n; s,r) \ ≤ (n-1) r/2 - s²/4; if |K| ≥ m+1, then max \(S) \; |\; S ∈ AKechel(m × n; r,r) \= r n - r(r+1)/2.