Richard Silverman’s translation enhances the accessibility and usability of Shilov’s Linear Algebra text for English readers, ensuring clarity and comprehension. Georgi E. Shilov, Professor of Mathematics at the Moscow State University, covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers. Because it contains an abundance of problems and examples, the book will be useful for self-study as well as for the classroom. Accessible and User-Friendly: Comprehensive Coverage: The book covers essential topics, including determinants, linear spaces, systems of linear equations, and coordinate transformations, providing a solid foundation for advanced studies. Clear Advanced Material: Starts with elementary content and seamlessly transitions into advanced areas, suitable for undergraduate and graduate courses. Abundant Practice Problems: Includes a plethora of problems with a dedicated section for hints and answers, facilitating self-study and reinforcing learning. Integrated Approach: Combines algebra, geometry, and analysis for a cohesive understanding of linear algebra concepts. Real-World Applications: Explores bilinear and quadratic forms, Euclidean and unitary spaces, and finite-dimensional algebras, providing practical and theoretical insights. Educationally Rich: Perfect for self-learners and classroom settings, offering extensive examples and exercises to enhance problem-solving skills. Appendix and Additional Resources: The appendix on categories of finite-dimensional spaces broadens the reader’s perspective and comprehension of the subject matter.

A thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second section addresses more advanced topics such as the study of canonical forms for matrices. The treatment can be tailored to satisfy the requirements of both introductory and advanced courses. Introductory courses that also serve as an initiation into formal mathematics will focus on the first six chapters. Students already schooled in matrices and linear mappings as well as theorem-proving will quickly proceed to selected chapters from part two. The selection can emphasize algebra or analysis/geometry, as needed. Ample examples, applications, and exercises appear throughout the text, which is supplemented by three helpful Appendixes.