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       Library of Congress Cataloging‐in‐Publication Data

      Names: O’Leary, Michael L., author.

      Title: Linear algebra / Michael L. O’Leary.

      Description: Hoboken, NJ : Wiley, 2021. | Includes bibliographical references and index.

      Identifiers: LCCN 2021008522 (print) | LCCN 2021008523 (ebook) | ISBN 9781119437444 (hardback) | ISBN 9781119437475 (adobe pdf) | ISBN 9781119437604 (epub) | ISBN 9781119437437 (obook)

      Subjects: LCSH: Algebras, Linear.

      Classification: LCC QA184 .O425 2021 (print) | LCC QA184 (ebook) | DDC 512/.5-dc23

      LC record available at https://lccn.loc.gov/2021008522

      LC ebook record available at https://lccn.loc.gov/2021008523

      Cover design by Wiley

      Cover image: © Hank Erdmann/Shutterstock

      This book is an introduction to linear algebra. Its goal is to develop the standard first topics of the subject. Although there are many computations in the sections, which is expected, the focus is on proving the results and learning how to do this. For this reason, the book starts with a chapter dedicated to basic logic, set theory, and proof‐writing. Although linear algebra has many important applications ranging from electrical circuitry and quantum mechanics to cryptography and computer gaming, these topics will need to wait for another day. The goal here is to master the mathematics so that one is ready for a second course in the subject, either abstract or applied. This may go against current trends in mathematics education, but if any mathematical subject can stand onits own and be learned for its own sake, it is the amazing and beautiful linear algebra.

      In addition to the focus on proofs, linear transformations play a central role. For this reason, functions are introduced early, and once the important sets of ℝn are defined in the second chapter, linear transformations are described in the third chapter and motivate the introduction of matrices and their operations. From there, invertible linear transformations and invertible matrices are encountered in the fourth chapter followed by a complete generalization of all previous topics in the fifth with the definition of abstract vector spaces. Geometries are added to the abstractions in the sixth chapter, and the book concludes with nice matrix representations. Therefore, the book’s structure is as follows.

      Euclidean Space The definition of ℝn is the focus of the second chapter with the main interpretation being that of arrows originating atthe origin. Euclidean distance and length are defined, and these are followed by the dot and cross products. Applications include planes and lines, areas and volumes, and the orthogonal projection.

      Transformations and Matrices Now that functions have been defined and interesting sets to serve as their domains and codomains have been given, linear transformations are introduced. After some basic properties, it is shown that these functions have nice representations as matrices. The matrix operations come next, their definitions being motivated by the definitions of the function operations. Linear operators on ℝ2 and ℝ3 serve as important examples of linear transformations. These include the reflections, rotations, contractions, dilations, and shears. The introduction of the kernel and the range is next. Issues with finding these sets motivate the need for easier techniques. Thus, Gauss–Jordan elimination and Gaussian elimination finally make their appearance.

      Invertibility The fourth chapter introduces the idea of an invertible matrix and ties it to the invertible linear operator. The standard procedure of how to find an inverse is given using elementary matrices, and inverses are then used to solve certain systems of linear equations. The determinant with its basic properties is next. How the elementary row operations affect the determinant is explained and carefully proved using mathematical induction. The next section combines the inverse and the determinant, and important results concerning both are proved. The chapter concludes with some mathematical applications including orthogonal matrices, Cramer’s Rule, and how the determinant can be used to compute the area or volume of the image of a polygon or a solid under a linear transformation.

      Inner Product Spaces Although ℝn is usually viewed as Cartesian space, it is technically just a set of n × 1 matrices. Any geometry that it has was given to it in the second chapter, even though its geometry is a copy of the geometry of Cartesian space. A close examination reveals that the geometry of ℝn is based on the dot product. Mimicking this, an abstract vector space is given its geometry with an inner product, which is a function defined so that it has the same basic properties as the dot product. The vector space then becomes an inner product space so that distances, lengths, and angles can be found using objects like matrices, polynomials, and functions. Other topics related to the inner product include a generalization of the orthogonal projection, orthonormal bases, direct sums, and the Gram–Schmidt process.

      Matrix Theory The book concludes with an introduction to the powerful concepts of eigenvalues and eigenvectors. Both the characteristic polynomial and the minimal polynomial are defined and used throughout the chapter. Generalized eigenvectors are presented and used to write ℝn as a direct sum of subspaces. The concept of similar matrices is given, and if a matrix does not have enough eigenvectors, it is proved that such matrices are similar to matrices with a nice form. This is where Schur’s Lemma makes its

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