This book discusses four major topics: systems of linear equations, eigenvalues and eigenvectors, the least squares approximation and linear programming. The main emphasis has been given to the presentation of a wide range of algorithms dealing with the problems mentioned above and their implementation on computers. The introductory chapter briefly explains computer representation of numbers (floating-point numbers) and the effect of round-off errors on basic arithmetic operations. The exposition proceeds with direct and indirect methods for solving linear systems (including a discussion of errors and conditioning), algorithms for the computation of eigenvalues and eigenvectors (e.g. the power method, the Jacobi method and QR and LR decomposition methods), the Aitken interpolation method and various forms of the least squares approximation (linear as well as nonlinear). The final chapter is devoted to linear programming, especially to the simplex method and duality. The author assumes no previous knowledge of linear algebra. Most theorems are stated without proof; on the other side, there are many carefully worked-out examples demonstrating all the algorithms in action and additional exercises. The algorithms are implemented in Matlab; the appendix provides a readable introduction to this system. The book could be useful even for readers who do not use Matlab.