Section Description

Linear algebra deals with the solution of systems of linear equations and the analysis of linear systems (roughly speaking, systems in which the output is proportional to the input). Both sorts of systems are often represented by tables of numbers called matrices (singular: matrix). Consequently, much of beginning linear algebra centers around matrix manipulation. It turns out that, if one is careful, one can often treat a matrix---a table of numbers---as if it were a special kind of `number? in its own right. A good part of linear algebra also involves learning how to use geometric thinking to get insight into algebraic and numerical problems. What is surprising is that the spatial intuition we have picked up merely from living can often be applied---again, if we are careful---to understand problems that seem to involve nothing but numbers. The course covers these topics: basic solution methods for systems of linear equations, with applications (e.g., balancing chemical equations, networks, economic models); basic vectors (`geometric? thinking); matrix algebra, including matrix inverses and factorization; abstract vector spaces (more `geometry?); determinants; eigenvalues and eigenvectors, with applications (e.g., dynamical systems, systems of differential equations). Depending on class or instructor preference, and on time, the course can also include treatment of least-squares problems and the singular value decomposition (with applications).