M  A  T  H 
2  1  B 
The number of row reduced n x n matrices is given by the sequence
A006116, a sum of Gaussian Binomial coefficients.
It is the number of subspaces of a vector space over the field Z2
or the number of distinct binary linear codes of length n
or the number of Abelian groups in C_{2}^{n}.
The sequence starts as follows:
1, 2, 5, 16, 67, 374, 2825, 29212, 417199, 8283458, ...Here are all 15 nonzero 3x3 cases: (* How many row reduced 01 matrices are there of size n x n ? *) F[n_]:=Module[{i,u,m=n^2}, i[x_]:=Partition[PadLeft[IntegerDigits[x,2],m],n]; u=Map[i,Range[0,2^m1]]; (* produces list of all 01 matrices *) v=Union[Table[RowReduce[u[[k]]],{k,2^m1}]]; w={}; Do[If[Sort[Union[Flatten[v[[k]]]]]=={0,1}, w=Append[w,v[[k]]]],{k,Length[v]}]; 1+Length[w]]; F[4]When first computing this, I (Oliver) had not checked that the row reduced matrices are still 01 matrices. Actually, they are not in general. Thanks to Jun Hou Fou (whom you know from teaching 21a last semester) for pointing this out to me and telling about the connection to the subspace problem and the sequence A006116.
Please send questions and comments to knill@math.harvard.edu
Math21b Harvard College Class Number:16325 Course ID:110989 Oliver Knill  Spring 2016 
Department of Mathematics 
Faculty of Art and Sciences 
Harvard University,
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