Equivalent Conditions for Singular and Nonsingular Matrices Let a Be an N × N Matrix

Equivalent Conditions for Singular and Nonsingular Matrices Let a Be an N × N Matrix

Equivalent Conditions for Singular and Nonsingular Matrices Let A be an n × n matrix. Any pair of statements in the same column are equivalent. A is singular (A−1 does not exist). A is nonsingular (A−1 exists). Rank(A) = n. Rank(A) = n. |A| = 0. |A| = 0. A is not row equivalent to In. A is row equivalent to In. AX = O has a nontrivial solution for X. AX = O has only the trivial solution for X. AX = B does not have a unique AX = B has a unique solution solution (no solutions or for X (namely, X = A−1B). infinitely many solutions). The rows of A do not form a The rows of A form a basis for Rn. basis for Rn. The columns of A do not form The columns of A form abasisforRn. abasisforRn. The linear operator L: Rn → Rn The linear operator L: Rn → Rn given by L(X) = AX is given by L(X) = AX not an isomorphism. is an isomorphism. Diagonalization Method To diagonalize (if possible) an n × n matrix A: Step 1: Calculate pA(x) = |xIn − A|. Step 2: Find all real roots of pA(x) (that is, all real solutions to pA(x) = 0). These are the eigenvalues λ1, λ2, λ3, ..., λk for A. Step 3: For each eigenvalue λm in turn: Row reduce the augmented matrix [λmIn − A | 0] . Use the result to obtain a set of particular solutions of the homogeneous system (λmIn − A)X = 0 by setting each independent variable in turn equal to 1 and all other independent variables equal to 0. Step 4: If after repeating Step 3 for each eigenvalue, you have less than n fundamental eigenvectors overall for A,thenA cannot be diagonalized. Stop. Step 5: Otherwise, form a matrix P whose columns are these n fundamental eigenvectors. −1 Step 6: Verify that D = P AP is a diagonal matrix whose dii entry is the eigenvalue for the fundamental eigenvector forming the ith column of P.AlsonotethatA = PDP−1. Simplified Span Method (Simplifying Span(S)) Suppose that S is a finite subset of Rn containing k vectors, with k ≥ 2. To find a simplified form for span(S), perform the following steps: Step 1: Form a k × n matrix A by using the vectors in S as the rows of A. (Thus, span(S) is the row space of A.) Step 2: Let C be the reduced row echelon form matrix for A. Step 3: Then, a simplified form for span(S) is given by the set of all linear combinations of the nonzero rows of C. Independence Test Method (Testing for Linear Independence of S) Let S be a finite nonempty set of vectors in Rn. To determine whether S is linearly independent, perform the following steps: Step 1: Create the matrix A whose columns are the vectors in S. Step 2: Find B, the reduced row echelon form of A. Step 3: If there is a pivot in every column of B,thenS is linearly independent. Otherwise, S is linearly dependent. Coordinatization Method (Coordinatizing v with Respect to an Ordered Basis B) n Let V be a nontrivial subspace of R ,letB = (v1, ..., vk) be an ordered basis for V, n and let v ∈ R .Tocalculate[v]B, if it exists, perform the following: Step 1: Form an augmented matrix [A | v] by using the vectors in B as the columns of A, in order, and using v as a column on the right. Step 2: Row reduce [A | v] to obtain the reduced row echelon form [C | w]. Step 3: If there is a row of [C | w] that contains all zeroes on the left and has a nonzero entry on the right, then v ∈/ span(B) = V, and coordinatization is not possible. Stop. Step 4: Otherwise, v ∈ span(B) = V. Eliminate all rows consisting entirely of zeroes in [C | w] to obtain [Ik | y]. Then, [v]B = y, the last column of [Ik | y]. Transition Matrix Method (Calculating a Transition Matrix from B to C) To find the transition matrix P from B to C where B and C are ordered bases for a nontrivial k-dimensional subspace of Rn, use row reduction on 1st 2nd kth 1st 2nd kth ⎡ vector vector ··· vector vector vector ··· vector ⎤ in in in in in in ⎢ CC C BB B⎥ ⎣ ⎦ to produce I P k . rows of zeroes Elementary Linear Algebra Elementary Linear Algebra Fifth Edition Stephen Andrilli Department of Mathematics and Computer Science La Salle University Philadelphia, PA David Hecker Department of Mathematics Saint Joseph’s University Philadelphia, PA AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 125 London Wall, London, EC2Y 5AS, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright © 2016, 2010, 1999 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-12-800853-9 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress For information on all Academic Press publications visit our website at http://store.elsevier.com/ To our wives, Ene and Lyn, for all their help and encouragement Preface for the Instructor This textbook is intended for a sophomore- or junior-level introductory course in linear algebra. We assume the students have had at least one course in calculus. PHILOSOPHY OF THE TEXT Helpful Transition from Computation to Theory: Our main objective in writing this textbook was to present the basic concepts of linear algebra as clearly as possible. The “heart” of this text is the material in Chapters 4 and 5 (vector spaces, linear transformations). In particular, we have taken special care to guide students through these chapters as the emphasis changes from computation to abstract theory. Many theoretical concepts (such as linear combinations of vectors, the row space of a matrix, and eigenvalues and eigenvectors) are first introduced in the early chapters in order to facilitate a smoother transition to later chapters. Please encourage the students to read the text deeply and thoroughly. Applications of Linear Algebra and Numerical Techniques: This text contains a wide variety of applications of linear algebra, as well as all of the standard numerical techniques typically found in most introductory linear algebra texts. Aside from the many applications and techniques already presented in the first seven chapters, Chapter 8 is devoted entirely to additional applications, while Chapter 9 introduces several other numerical techniques. A summary of these applications and techniques is given in the chart located at the end of the Prefaces. Numerous Examples and Exercises: There are 340 numbered examples in the text, at least one for each major concept or application, as well as for almost every theorem. The text also contains an unusually large number of exercises. There are more than 970 numbered exercises, and many of these have multiple parts, for a total of more than 2600 questions. The exercises within each section are generally ordered by increasing difficulty, beginning with basic computational problems and moving on to more theoretical problems and proofs. Answers are provided at the end of the book for approximately half of the computational exercises; these problems are marked with a star (⋆). Full solutions to these starred exercises appear in the Student Solutions Manual. The last exercises in each section are True/False questions (there are over 500 of these altogether). These are designed to test the students’ understanding of fundamental concepts by emphasizing the importance of critical words in definitions or theorems. Finally, there is a set of comprehensive Review Exercises at the end of each of Chapters 1 through 7. Assistance in the Reading and Writing of Mathematical Proofs: To prepare students for an increasing emphasis on abstract concepts, we introduce them to proof-reading and xi xii Preface for the Instructor proof-writing very early in the text, beginning with Section 1.3,whichisdevoted solely to this topic.

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