A concise introduction to linear algebra

Géza Schay A Concise Introduction to Linear Algebra Department of Mathematics Géza Schay University of Massachusetts Boston, MA, USA Mathematics Subject Classification (2010) 15Axx Library of Congress[.]

Géza Schay A Concise Introduction to Linear Algebra Géza Schay Department of Mathematics University of Massachusetts Boston, MA, USA ISBN 978-0-8176-8324-5 978-0-8176-8325-2 (eBook) DOI 10.1007/978-0-8176-8325-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2012934162 Mathematics Subject Classification (2010): 15Axx MATLAB® is a registered trademark of The MathWorks, Inc For MATLAB and Simulink product information, please contact: The MathWorks, Inc Apple Hill Drive, Natick, MA, 01760-2098 USA Tel: 508-647-7000 Fax: 508-647-7001 E-mail: info@mathworks.com Web: mathworks.com This book is based on the author’s Introduction to Linear Algebra, published by Jones & Bartlett in 1996 © Springer Science+Business Media, LLC 2012 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhäuser-science.com) Index Multilinear function, 223 Multiplicity, 257 n-dimensional Euclidean vector space, 11 Natural permutation, 226 Nonparametric equations for a line, 29 Nonsingular matrix, 93 Nontrivial solution, 59 Norm, 16 Normal equations, 200 Normal vector, 34 Normalization, 212 Null space, 108, 124 Nullity, 133 Oblique coordinates, 26 Ohm’s law, 62 One-to-one mapping, 175 Onto mapping, 175 Order of matrix, 87 Ordered basis, 148 Orthogonal complement, 136 Orthogonal decomposition, 211 Orthogonal diagonalization, 274 Orthogonal matrix, 213 Orthogonal projection, 21 Orthogonal subspaces, 136 Orthogonal vectors, 17 Orthographic projection, 193 Orthonormal, 24 Orthonormal decomposition, 212 Orthonormal set of vectors, 212 Outer product, 75 Overdetermined, 48 Parallelogram law, Parametric equations of a line, 28 Parametric equations of a plane, 32 Partial pivoting, 302 Particular solution, 58 Permutation, 226 Permutation matrix, 97 Pivot, 46, 53 Position vector, Power method for eigenvalues, 306 Principal Axis Theorem, 274 Projection, 21, 200, 212 Projection matrix, 195, 203 Proper subspaces, 106 Pseudoinverse, 209 QR factorization, 216 Quadratic form, 275 329 Radius vector, Range, 175 Rank, 55, 177 Rank deficient, 56 Reduced echelon matrix, 60, 121 Reduction formulas, 235 Right inverse, 96 Right-handed coordinate system, 11 Rotation matrix, 67 Row rank, 133 Row space, 122 Row vector, 44 Row-equivalent matrices, 47 Scalar multiple of a mapping, 68 Scalar multiple of a matrix, 69 Scalar product, 17 Scalars, 2, 280 Scaled partial pivoting, 302 Schră odingers equation, 288 Schur’s lemma, 278 Similar matrices, 157 Similarity transformation, 157 Singular matrix, 93 Skew-symmetric matrix, 233 Span, 108 Spectral Theorem, 284 Spectral Theorem for Symmetric Matrices, 274 Square matrix – order of, 87 Standard basis, 118 Standard vectors, 23 Submatrix, 54 Subspace, 105 Sum of mappings, 68 Sum of matrices, 69 Sum of subspaces, 136 Symmetric matrix, 203, 273 Tensor product, 172 Tetrahedron, 14 Three-dimensional Euclidean vector space, Trace, 160 Transformation, 163 Transition matrix, 149 Transpose of a matrix, 94 Transpose of a product, 94 Transpose of the inverse, 94 Transposition, 226 Transversal, 36 330 Index Triangle inequality, 16 Triangle law, Triple product, 252 Trivial solution, 59 Trivial subspaces, 106 Two-dimensional Euclidean vector space, Underdetermined, 48 Unit matrix, 82 Unit vector, 16 Unitary matrix, 282 Upper triangular matrix, 56 Vandermonde determinant, 234 Vector product, 245 Vector properties, Vector space, 99 – of linear transformations, 181 Viewplane, 193 Work, 23 Zero matrix, 81 ... Géza Schay A Concise Introduction to Linear Algebra Géza Schay Department of Mathematics University of Massachusetts Boston, MA, USA ISBN 978-0-8176-8324-5 978-0-8176-8325-2... n-dimensional Euclidean vector space, 11 Natural permutation, 226 Nonparametric equations for a line, 29 Nonsingular matrix, 93 Nontrivial solution, 59 Norm, 16 Normal equations, 200 Normal vector,... Orthonormal decomposition, 212 Orthonormal set of vectors, 212 Outer product, 75 Overdetermined, 48 Parallelogram law, Parametric equations of a line, 28 Parametric equations of a plane, 32 Partial