I have a linear system of equations of size mxm, where m is large.  However, the variables that I'm interested in are just the first n variables (n is small compared to m).  Is there a way I can approximate the solution for the first m values without having to solve the entire system?  If so, would this approximation be faster than solving the full linear system?  
Can the solution of a linear system of equations be approximated for only the first few variables?
linear algebra;approximation algorithms
As others have pointed out, this is difficult to do with a direct solver. That said, it isn't that hard to do with iterative solvers. To this end, note that most iterative solvers in one way or another minimize the error with respect to some norm. Oftentimes, this norm is either induced by the matrix itself, but sometimes it is also just the l2 vector norm. But that doesn't have to be the case: you can choose which norm you want to minimize the error (or residual) in, and you could, for example, choose a norm in which you weigh the components you care about with 1 and all others with 1e-12, i.e. for example something like  $|| x ||^2  =  \sum_{i=1}^5 x_i^2 + $(1e-24)$\sum_{i=6}^N x_i^2$and corresponding scalar product. Then write all steps of the iterative solver with respectto this norm and scalar product, and you get an iterative solver that pays significantlymore attention to the vector elements you care about than to the others.The question of course is whether you need fewer iterations than with the norm/scalar product that weighs all components equally. But that should indeed be the case: let'ssay you only care about the five first vector elements. Then you should need at mostfive iterations to reduce the error by a factor of 1e12 since five iterations is whatis needed for the 5x5 system that describes them. That's not a proof but I'm pretty certain that you should indeed get away with a far smaller number of iterations if the weight in the norm (1e-12 above) is smaller than the tolerance with which you want to solve the linear system iteratively.