Why in this 21st century, solving Ax=b fast is still of importance?
The reasons are simple. Firstly I hope you know that
Ax=b is not just popping out from the usually `boring'
linear algebra lectures. It comes mainly from solving other
hard equations such as differential or integral equations.
Secondly high resolutions demanded by computer
solution of practical problems often require more than
N0=1,000,000 variables (say in oceanography or even in digital image
restoration, N=10*N0 is quite normal). Do you know that even with the
fastest computer getting the solution of a linear system
of such a large size may take at least 10 years (depending
if the matrix has any convenient zeros that can be avoided in
calculations)?
Thus the practical answer to the question is simply: how many 10 years
does an average researcher have in his or her life time?
Do we wait for existing mathematical methods to get the answer
for us or get our hands dirty in inventing fast methods?
Computational mathematicians are doing a lot of the latter.