For linear system Ax = b, the traditional condition number is the worst case for all b¡¦s and often overestimated in many problems. For a specific b, the effective condition number is a better upper bound for the relative error of x. But, it is also possible that this effective condition number is overestimated. In this thesis, we study the true ratio of the relative error of x to the relative perturbation of b, called the true condition number. We obtain several new upper bounds and estimates for true condition number. We also explore to change the system to an equivalent one by shifting b to minimize its effective condition number. Finally we apply all our results to functional approximation.