This textbook is designed for a first course in linear algebra for undergraduate students from a wide range of quantitative and data driven fields. By focusing on applications and implementation, students will be prepared to go on to apply the power of linear algebra in their own discipline. With an ever-increasing need to understand and solve real problems, this text aims to provide a growing and diverse group of students with an applied linear algebra toolkit they can use to successfully grapple with the complex world and the challenging problems that lie ahead. Applications such as least squares problems, information retrieval, linear regression, Markov processes, finding connections in networks, and more, are introduced on a small scale as early as possible and then explored in more generality as projects. Additionally, the book draws on the geometry of vectors and matrices as the basis for the mathematics, with the concept of orthogonality taking center stage. Important matrix factorizations as well as the concepts of eigenvalues and eigenvectors emerge organically from the interplay between matrix computations and geometry.
The R files are extra and freely available. They include basic code and templates for many of the in-text examples, most of the projects, and solutions to selected exercises. As much as possible, data sets and matrix entries are included in the files, thus reducing the amount of manual data entry required.
Introduction.-
1. Vectors.-
2. Matrices.-
3. Matrix Contexts.-
4. Linear Systems.-
5. Least Squares and Matrix Geometry.
6. Orthogonal Systems.-
7. Eigenvalues.-
8. Markov Processes.-
9. Symmetric Matrices.-
10. Singular Value Decomposition.-
11. Function Spaces.-Bibliography.-Index.
Timothy G. Feeman is professor of mathematics, Villanova University, in Lancaster, Pennsylvania. His original area of research is the theory of operators on Hilbert spaces once described as "the field of mathematics that has the strongest interaction with the scientific and technological developments which are characteristic of the twentieth century." Since the mid- to late-1990s, his scholarly efforts have become more diversified. Professor Feeman is the author of The Mathematics of Medical Imaging, also published in the "Springer Undergraduate Texts in Mathematics and Technology" series.
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