About this Book
Elementary Linear Algebra, 5th edition, by Stephen Andrilli and David Hecker, is a textbook for a beginning course in linear algebra for sophomore or junior mathematics majors. This text provides a solid introduction to both the computational and theoretical aspects of linear algebra.
Chapters 1 through 3 introduce students to vectors, matrices, linear systems, matrix inverses, determinants, eigenvalues and eigenvectors. Chapters 4 through 6 introduce real vector spaces, span, linear independence, basis and dimension, coordinatization, linear transformations, isomorphisms, a deeper look at eigenvalues and eigenvectors, and orthogonality. Chapter 7 is devoted to extending the results of earlier chapters to complex vector spaces.
Numerous real-world applications of linear algebra are presented in the text. In particular, an entire chapter (Chapter 8) is devoted solely to applications, including such diverse areas as graph theory, circuit theory, Markov chains, elementary coding theory, least-squares polynomials and least-squares solutions for inconsistent systems, differential equations, computer graphics and quadratic forms.
Many common computational techniques in linear algebra are covered, such as Gauss-Jordan row reduction, calculating determinants by both cofactor expansion and row reduction, Cramer’s Rule, determining whether sets of vectors span a vector space or are linear independent, constructing bases as well as finding the transition matrix between bases, calculating the matrix of a linear transformation as well as its kernel and range, determining whether a linear transformation is one-to-one or onto, the Gram-Schmidt Process, orthogonal projection onto a subspace, and diagonalization as well as orthogonal diagonalization of a matrix. An entire chapter (Chapter 9) is completely devoted to further computational methods: iterative methods (Jacobi and Gauss-Seidel) for solving linear systems, LDU Decomposition, the Power Method for finding eigenvalues, QR Decomposition, and concluding with Singular Value Decomposition and its usefulness in digital imaging.
The most unique feature of the text is that students are nurtured in the art of creating mathematical proofs using linear algebra as the underlying context. An early section (Section 1.3) introduces students to several important types of proof methods and related considerations that they need to keep in mind. For virtually all of the major theorems in the text, either a proof is given in a style for students to study (thereby improving their proof-reading skills), or else an outline or major hint is presented from which the students can complete the proof themselves.
The text contains a large number of worked out examples, as well as more than 970 exercises (with over 2600 total questions) to give students practice in both the computational aspects of the course and in developing their proof-writing abilities. Every section of the text ends with a series of true/false questions carefully designed to test the students’ understanding of the material. In addition, each of the first seven chapters concludes with a thorough set of review exercises and additional true/false questions.
Supplements to the text include an Instructor’s Manual with answers to all of the exercises in the text, and a Student Solutions Manual with detailed answers to the starred exercises in the text. Both of these supplements were completely written by the authors themselves so that they would conform as closely as possible to the style of the text.
Finally, there are seven web sections available on the book’s website to instructors who adopt the text for use with their students. These web sections contain additional material for further study: lines and planes and the cross product in 3-space, function spaces, the Jacobian and Hessian matrices, Jordan Canonical Form, first-order systems of linear homogeneous differential equations, and isometries on inner product spaces. A corresponding Instructor’s Manual and Student Solutions Manual are also available for these web sections as well.