I'm looking for the fastest algorithm/package i could use to compute the null space of an extremely large (millions of elements, and not necessarily square) matrix. Any language would be alright, preferably something in Python/C/C++/Java. Your help would be greatly appreciated!
In what kind of data structure is your matrix represented? 
If you use an element list to represent the matrix, i.e. "column, row, value" tuple for one matrix element, then the solution would be just count the number of the tuples (subtracted by the matrix size)
Computing the null space of a large matrix