Generating a d-dimensional linear subspace efficiently

We present an algorithm for computing a d-dimensional subspace of the row space of a matrix. For an n x n matrix A with m nonzero entries and with rank(A) ≥ d the algorithm generates a d x n matrix with full row rank and which is a subspace of Rows(A). If rank(A) < d the algorithm generates a rank(A) x n row-equivalent matrix. The running time of the algorithm is O(min{n ^2-2/w m^1/w d^w-2+1/w, n² d ^w-2}) where w < 2.376 is the matrix multiplication exponent. An immediate corollary of the algorithm is the construction of a row-reduced equivalent matrix of A, and hence the computation of rank(A), in time O(min{n^2-2/w m^1/w rank(A)^w-2+1/w, n² rank(A)^w-2}). We note that the running time is sub-quadratic if d < (n²/m)^0.528.