How to choose rank(A) independant columns of matrix A efficiently

Question

Is there a better way than brute forcing (choose $k=\mathrm{rank}(A)$ first columns - test the determinant, if determinant = 0 choose new column set - there are $\binom nk$ many possibilities which is inefficient if the full rank sub matrices are rare!)

Answer 1

Compute the row echelon form of the matrix and select the pivot columns. Computing the row echelon form of a $m\times n$ matrix will take $O(m^2n)$ field operation, which is pretty straightforward. If full rank submatrices are not sparse, you can save further work by ignoring some of the columns.