MCS 221: Linear Algebra

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Linear algebra is a branch of mathematics that is remarkable for its breathtaking beauty and ubiquitous utility. It provides a compact, expressive language for the description of models arising in a wide variety of fields of study, including biology, chemistry, computer science, data transmission, ecology, economics, engineering, image processing, medicine, physics, psychology, sociology, and statistics, as well as other areas of mathematics. Its study typically is a requirement for students of engineering, science, and mathematics.

The fundamental theme of linear algebra is the solution of systems of linear equations, like

2 x + 3 y = 4 and 5 x - 6 y = 7.

The equations are called "linear" because geometrically they describe lines (or planes or higher-dimensional counterparts). The central equations of linear algebra, in matrix notation, are the linear system A x = b and the eigenvalue problem A x = m x. At any given time, a significant fraction of the planet's computing resources is involved in solving such problems.

The fundamental concept of linear algebra is the vector space. This notion embraces lines, planes, and Euclidean 3-dimensional space. But vector spaces include much, much more. The student who masters this abstraction will obtain a key to understanding a wide variety of problems. Another key concept is that of a linear transformation between vector spaces. This abstract concept has concrete representations in the form of rectangular arrays of numbers called matrices.

MCS 221 students will be encouraged to understand the fundamental ideas of linear algebra the GAC way: geometrically, algebraically, and computationally. They will apply their understanding both to "problems to solve" and to "problems to prove." Through a variety of applications, students will gain an appreciation of the widespread usefulness of linear algebra. In addition, students will find themselves seduced by the beauty of the subject as they gain experience in mathematical abstraction and logical deduction.