A sparse matrix is a matrix with very few nonzero elements. Many applications in diverse fields give rise to linear systems of the form $Ax = b$, where $A$ is sparse. The problem in solving these systems is to take advantage of the preponderance of zero elements to reduce both memory use and comutation time. The purpose of this paper is to introduce students (and perhaps their teachers) to sparse matrix technology. It is impossible to treat all the techniques developed since the subject started in the 1960's. Instead, this paper constructs a sparse solver for positive definite systems that would have been state of the art around 1980, emphasizing equally theory and computational practice. It is hoped that a mastery of this material will allow the reader to study the subject independently. (UMIACS-TR-2003-95)