We propose an approach that provides a stable transformation of the output of a linear system into the output of a system with the desired basis. The matrix of basis functions of the linear system has a large condition number, and the series of its singular numbers gradually decreases to zero. Two types of methods for stable output transformation are developed using the approximation of matrices based on the truncated Singular Value Decomposition and on the Random Projection with different types of random matrices. It is shown that the use of the output transformation as preprocessing increases the accuracy of solving sparse approximation problems. An example of using the method to determine the activity of weak radiation sources is considered.