<p>This paper first gives rigorous estimates for the convergence rates of the randomized extended Kaczmarz (REK) method and its variants when they are applied to solve large sparse linear systems. The REK method approximates the least-norm least-squares solution by introducing an auxiliary vector. However, in some cases, its convergence rates is deteriorated due to repeated selection of certain specific columns. To overcome this drawback, we modify the REK method and propose a REK method with double-space residuals (REKDR). In the selections of the working rows and columns, the REKDR method adopts a random selection rule based on the probability of their related residuals, so its convergence rate could be improved significantly. In addition, we analyze the convergence property of the REKDR method, and examine its effectiveness by numerical experiments. The experimental results show that REKDR method has faster convergence rate and smaller computing time than the existing methods in this class when they are employed to solve inconsistent linear systems.</p>