Linear Algebra II

These volumes introduce and develop the theory of linear structures for students of mathematics and its applications. Today, linear algebra serves as an essential tool and unifying language across nearly all areas of mathematics. Given its importance in the natural sciences, engineering, and economics, linear algebra is presented as a valuable and widely applicable subject in its own right.

In addition to the standard core material, the volume includes topics of interest to students from a variety of disciplines. Students of mathematics education will find an introduction to several aspects of analytic geometry. In this second volume, the study of analytic geometry is extended through a treatment of quadrics and, in particular, the theory of polyhedra, culminating in linear optimization and the simplex algorithm. Readers with an interest in algebra are introduced to bi- and multilinear forms and multilinear algebra, while those interested in analysis or physics will find a thorough treatment of spectral theory and linear ordinary differential equations. The Schur and Jordan normal forms, including their real counterparts, are fundamental to the development of the theory.

Throughout, structural insights are combined with practical algorithmic methods. The treatment of tensor calculus helps bridge the gap between the mathematical and engineering perspectives, while convex geometry facilitates the transition to functional analysis and its applications. Students pursuing numerical mathematics, optimization, or data-oriented applications will encounter topics such as singular value decomposition, principal component analysis, and linear and quadratic optimization. Numerical methods are further developed through the study of least-squares problems and QR decomposition, providing a foundation for the modern eigenvalue algorithms described in Volume 3. Many of these topics are also of special relevance to students of economics and related disciplines.

Particular emphasis is placed on connecting theory and algorithms and on relating both to applications in the sciences. To support this goal, mathematical modelling plays a central role. Ongoing examples drawn from mechanics, electrical networks, and economics are developed progressively alongside the theory.