Introduction to Linear Algebra / Edition 2by Serge Lang
Pub. Date: 03/01/1997
Publisher: Springer New York
This is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues. The book contains a large number of exercises, some of the routine computational type, while others are conceptual.
Table of ContentsI Vectors.- II Matrices and Linear Equations.- III Vector Spaces.- IV Linear Mappings.- V Composition and Inverse Mappings.- VI Scalar Products and Orthogonality.- VII Determinants.- VIII Eigenvectors and Eigenvalues.- Answers to Exercises.
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This text, parts of which are drawn from Lang's Calculus of Several Variables and his more theoretical text Linear Algebra, provides a solid introduction to the subject for students of pure mathematics. The text covers vectors, matrices and systems of linear equations, vectors transformations, linear mappings, composition and inverse mappings, scalar products and orthogonality, determinants, and eigenvectors and eigenvalues. Lang demonstrates computational techniques through clearly written examples and carefully develops the basic theory. Since the emphasis is on the development of the theory, the text is best suited to students of pure mathematics. The exposition is generally clear, but at times I had to refer to the texts Beginning Linear Algebra by Blyth and Robertson and Linear Algebra by Friedberg, Insel, and Spence for clarification. The exercises include both computational problems, which require meticulous attention to detail and can sometimes be tedious, and proofs of results that extend the material developed in the text. Some problems are reintroduced after new material has been developed, enabling you to solve the problems in new ways once you have additional tools at your disposal. Answers to most of the exercises are provided in an appendix, making the text suitable for self-study. Since this text was designed for a one semester class for students who have not necessarily completed multi-variable calculus, its scope is limited. Students preparing to do graduate work in mathematics will need to read an additional text such as Lang's Linear Algebra, Friedberg, Insel, and Spence's Linear Algebra, Hoffman and Kunze's Linear Algebra, Axler's Linear Algebra Done Right, or Blyth and Robertson's Further Linear Algebra.