Linear Algebra Software

The cybernetic revolution that occurred on the eve of the twenty–first century affected our life in all demeanors. The collection, organization, transformation, and interpretation of information are the cornerstones of science, industry, business, and government. Each of these aspects of information management uses mathematics as a tool.

Linear algebra has undergone a real metamorphosis during this century, partly due to the birth and development of computers. It is so active that entire periodical magazines are now devoted to it, and one single tutorial can only reflect part of its vitality. Here is an attempt to face this challenge in a concise—although rigorous—manner. Linear algebra is a general and powerful language. This tutorial is based on utilization of powerful software packages and focuses on applications, justifying the elaboration of such an abstract language.

Linear algebra is employed in most computational physics and information technologies. The two main problems that linear algebra studies can be trivially stated as follows:

• Solve a linear system of m equations with n unknowns: Ax = b.
• Determine a spectral decomposition of a matrix A → Ax = λx.

Although solutions to these two problems are well known, entire monographs have been dedicated to their efficient solutions and numerical implementations. This website is designed to enhance your knowledge of linear algebra, a core component of computer science and various engineering fields. While it is geared toward individuals with prior education in algebra, some symbols or terminology may be unfamiliar. We aim to provide clear explanations throughout; however, if anything remains unclear, we encourage you to consult a search engine or AI assistant for additional support.

The main software packages we recommend are:

• Matlab® (Octave)
• Maple™
• Mathematica® (Mathics)
• R
• Maxima
• Sage
• SymPy
• Python
• Julia

All these packages have syntax commands closely related to traditional pen-and-paper math.