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The Third Edition of A Portrait of Linear Algebra builds on the strengths of the previous editions, providing the student a unified, elegant, modern, and comprehensive introduction: • emphasizes the reading, understanding, and writing of proofs, and gives students advice on how to master these skills; • presents a thorough introduction to basic logic, set theory, axioms, theorems, and methods of proof; • develops the properties of vector and matrix operations as natural extensions of the field axioms for real numbers; • gives an early introduction of the core concepts of spanning, linear independence, subspaces (including the fundamental matrix spaces and orthogonal complements), basis, dimension, kernel, and range; • explores linear transformations and their properties by using their correspondence with matrices, fully investigating injective, surjective, and bijective transformations; • focuses on the derivative as the prime example of a linear transformation on function spaces, establishing the strong connection between the fields of Linear Algebra and Differential Equations; • comprehensively introduces infinite cardinalities and infinitedimensional vector spaces; • thoroughly develops Permutation Theory to completely prove the properties of determinants; • presents large non-trivial matrices, especially symmetric matrices, that have multi-dimensional eigenspaces; • rigorously constructs Complex Euclidean Spaces and inner products, with complete proofs of Schur’s Lemma, the Spectral Theorems for normal matrices, and the simultaneous diagonalization of commuting normal matrices; • proves and applies the Fundamental Theorem of Linear Algebra, and its twin, the Singular Value Decomposition, an essential tool in modern computation; • presents application topics from Physics, Chemistry, Differential Equations, Geometry, Computer Graphics, Group Theory, Recursive Sequences, and Number Theory; • includes topics not usually seen in an introductory book, such as the exponential of a matrix, the intersection of two subspaces, the pre-image of a subspace, cosets, quotient spaces, and the Isomorphism Theorems of Emmy Noether, providing enough material for two full semesters; • features more than 500 additional Exercises since the 2nd Edition, including basic computations, assisted computations, true or false questions, mini-projects, and of course proofs, with multi-step proofs broken down with hints for the student; • written in a student-friendly style, with precisely stated definitions and theorems, making this book readable for selfstudy. The author received his Ph.D. in Mathematics from the California Institute of Technology in 1993, and since then has been a professor at Pasadena City College. COLORS: cyan magenta yellow black KH final proof: 6-24-16 jjf BOOK: 8.5x11 SPINE: 1.74 for Perfect Binding A Portrait of Linear Algebra Third Edition Jude Thaddeus Socrates Pasadena City College Kendall Hunt publishing c o mpany Jude Thaddeus Socrates and A Portrait of Linear Algebra are on Facebook. Please visit us! To order the print or e-book version of this book, go to: https://he.kendallhunt.com/product/portrait-linear-algebra Cover art: Linear Transformation by Jude Thaddeus Socrates, 2016 www.kendallhunt.com Send all inquiries to: 4050 Westmark Drive Dubuque, IA 52004-1840 Copyright © 2016 by Kendall Hunt Publishing Company ISBN 978-1-4652-9053-3 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright owner. Printed in the United States