The Less Is More Linear Algebra of Vector Spaces and Matrices

Designed for a proof-based course on linear algebra, this rigorous and concise textbook intentionally introduces vector spaces, inner products, and vector and matrix norms before Gaussian elimination and eigenvalues so students can quickly discover the singular value decomposition (SVD)―arguably the most enlightening and useful of all matrix factorizations. Gaussian elimination is then introduced after the SVD and the four fundamental subspaces and is presented in the context of vector spaces rather than as a computational recipe. This allows the authors to use linear independence, spanning sets and bases, and the four fundamental subspaces to explain and exploit Gaussian elimination and the LU factorization, as well as the solution of overdetermined linear systems in the least squares sense and eigenvalues and eigenvectors. This unique textbook also includes examples and problems focused on concepts rather than the mechanics of linear algebra. and The problems at the end of each chapter that and in an associated website encourage readers to explore how to use the notions introduced in the chapter in a variety of ways. Additional problems, quizzes, and exams will be posted on an accompanying website and updated regularly. The Less Is More Linear Algebra of Vector Spaces and Matrices is for students and researchers interested in learning linear algebra who have the mathematical maturity to appreciate abstract concepts that generalize intuitive ideas. The early introduction of the SVD makes the book particularly useful for those interested in using linear algebra in applications such as scientific computing and data science. It is appropriate for a first proof-based course in linear algebra.

Daniela Calvetti is the James Wood Williamson Professor in the Department of Mathematics, Applied Mathematics, and Statistics at Case Western Reserve University. Her research interests, strongly rooted in numerical analysis and numerical linear algebra, include inverse problems, uncertainty quantification, and mathematical modeling, particularly of human metabolism and the brain. She is the 2020-2022 program director of the SIAM Activity Group on Uncertainty Quantification. Erkki Somersalo is a professor in the Department of Mathematics, Applied Mathematics, and Statistics at Case Western Reserve University. With a background in analysis and partial differential equations, his main areas of interest include computational inverse problems, with an emphasis on Bayesian methods and applications to a wide range of areas, particularly medical imaging. He is working on mathematical modeling of complex biological systems. He is a founding member of the Finnish Inverse Problems Society, a member of the Finnish Academy of Sciences and Letters, and a fellow of the Institute of Physics, UK.