What is the asymptotically fastest known algorithm for computing the nullspace of a matrix? I know Gaussian Elimination takes $O(n^3)$ arithmetic operations, but I'm unsure if any better algorithms are known.
The exponent of computing a basis of the kernel is the same
as the exponent of matrix multiplication, see the 
book Algebraic Complexity Theory by Bürgisser, Clausen &amp; Shokrollahi.
So it can be done in time $O(n^{2.38})$.