I know the best algorithm to solve a linear system in $ \mathbb{R}$ with $ n$ variables is Coppersmith-Winograd’s algorithm, which has a complexity of $ $ O\left(n^{2.376}\right). $ $ How much easier is it to simply determine whether the same system has any solution?

More precisely, given a system of $ m$ equations and $ n$ unknowns, what is the complexity of establishing whether it has any solution?