Linear Algebra FOR DUMmIES ‰ Linear Algebra FOR DUMmIES ‰ by Mary Jane Sterling Linear Algebra For Dummies® Published by Wiley Publishing, Inc 111 River St Hoboken, NJ 07030-5774 www.wiley.com Copyright © 2009 by Wiley Publishing, Inc., Indianapolis, Indiana Published simultaneously in Canada 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, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http:// www.wiley.com/go/permissions Trademarks: Wiley, the Wiley Publishing logo, For Dummies, the Dummies Man logo, A Reference for the Rest of Us!, The Dummies Way, Dummies Daily, The Fun and Easy Way, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc and/ or its affiliates in the United States and other countries, and may not be used without written permission All other trademarks are the property of their respective owners Wiley Publishing, Inc., is not associated with any product or vendor mentioned in this book LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ For general information on our other products and services, please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993, or fax 317-572-4002 For technical support, please visit www.wiley.com/techsupport Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Control Number: 2009927342 ISBN: 978-0-470-43090-3 Manufactured in the United States of America 10 About the Author Mary Jane Sterling is the author of five other For Dummies titles (all published by Wiley): Algebra For Dummies, Algebra II For Dummies, Trigonometry For Dummies, Math Word Problems For Dummies, and Business Math For Dummies Mary Jane continues doing what she loves best: teaching mathematics As much fun as the For Dummies books are to write, it’s the interaction with students and colleagues that keeps her going Well, there’s also her husband, Ted; her children; Kiwanis; Heart of Illinois Aktion Club; fishing; and reading She likes to keep busy! Dedication I dedicate this book to friends and colleagues, past and present, at Bradley University Without their friendship, counsel, and support over these past 30 years, my teaching experience wouldn’t have been quite so special and my writing opportunities wouldn’t have been quite the same It’s been an interesting journey, and I thank all who have made it so Author’s Acknowledgments A big thank-you to Elizabeth Kuball, who has again agreed to see me through all the many victories and near-victories, trials and errors, misses and bull’s-eyes — all involved in creating this book Elizabeth does it all — project and copy editing Her keen eye and consistent commentary are so much appreciated Also, a big thank-you to my technical editor, John Haverhals I was especially pleased that he would agree to being sure that I got it right And, of course, a grateful thank-you to my acquisitions editor, Lindsay Lefevere, who found yet another interesting project for me Publisher’s Acknowledgments We’re proud of this book; please send us your comments through our Dummies online registration form located at http://dummies.custhelp.com For other comments, please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993, or fax 317-572-4002 Some of the people who helped bring this book to market include the following: Acquisitions, Editorial, and Media Development Project Editor: Elizabeth Kuball Acquisitions Editor: Lindsay Lefevere Copy Editor: Elizabeth Kuball Composition Services Project Coordinator: Patrick Redmond Layout and Graphics: Carl Byers, Reuben W Davis, Mark Pinto Assistant Editor: Erin Calligan Mooney Proofreaders: Laura L Bowman, Dwight Ramsey Editorial Program Coordinator: Joe Niesen Indexer: Christine Karpeles Technical Editor: John S Haverhals Senior Editorial Manager: Jennifer Ehrlich Editorial Supervisor and Reprint Editor: Carmen Krikorian Editorial Assistants: Jennette ElNaggar, David Lutton Cover Photos: PhotoAlto Agency Cartoons: Rich Tennant (www.the5thwave.com) Publishing and Editorial for Consumer Dummies Diane Graves Steele, Vice President and Publisher, Consumer Dummies Kristin Ferguson-Wagstaffe, Product Development Director, Consumer Dummies Ensley Eikenburg, Associate Publisher, Travel Kelly Regan, Editorial Director, Travel Publishing for Technology Dummies Andy Cummings, Vice President and Publisher, Dummies Technology/General User Composition Services Debbie Stailey, Director of Composition Services Index scalar multiplication, 108 scalar multiplication of matrices, 54 vector addition, 248–250 computers used for calculating matrices, 238 consecutive, 344 consistent system of equations defined, 344 overview, 66–67 constant defined, 344 matrices, 77 constraint, 344 contraction defined, 344 described, 25 of eigenvectors, 293 linear transformation, 168–169 of vectors, 25 coordinate axes, 344 coordinate plane vectors, 20–23 coordinate vector, 344 Cramer, Gabriel (mathematician), 233 Cramer’s rule applying, 232–234 assigning positions for, 231–232 described, 231 impossible statements, 235–236 infinite solutions, 235–236 •D• Dantzig, George Bernard (mathematician), 76 delta (∆), 340 dependent system of equations, 344 Derive, 238 determinants area of parallelogram, finding, 195–198 area of triangle, finding, 192–195 cofactor expansion, 189–192 columns, interchanging, 204–206 computing, 186–189 defined, 344 of identity matrix, 203–204 of inverse of a matrix, 203–204, 228–230 lower triangluar matrix, creating, 217–221 of manipulated matrices, 209–211, 212 of matrix products, 221–222 minor, 189–190 overview, 15–16, 185 parallelepipeds, finding volume of, 198–199 rows, interchanging, 204–206 × matrix, 187–189 transposing, 202–203 of triangular matrices, 213–214, 217–221 × matrix, 186–187 upper triangluar matrix, creating, 217–221 zero determinants, 206–209 diagonal matrices, 51, 304–307, 345 diagonalizable matrix, 345 dilation defined, 345 described, 25 of eigenvectors, 293 linear transformation, 168–169 of vectors, 25 dimension based on basis, 144–146 dimension of matrix, 43, 345 dimension of vector space, 345 distributive property addition of matrices, 55 defined, 345 linear transformation, 161 multiplication of matrices, 55 scalar multiplication, 108, 251–252 dividing matrices, 59–60 dot product (inner product) of two vectors, 24, 35–37, 345 •E• economic matrix, 321–322 eigenvalues defined, 291, 345 of diagonal matrices, 305–307 of matrix inverse, 301–302 of matrix transpose, 300 overview, 16–17 of powers of square matrices, 303–304 solving for, 294–299 353 354 Linear Algebra For Dummies eigenvalues (continued) × matrix, 297–299 of triangular matrix, 302–303 × matrix, 294–296 eigenvectors contracting, 293 defined, 291, 345 of diagonal matrices, 305–307 dilating, 293 example, 290 of matrix inverse, 301–302 overview, 16–17 rotating, 291–292 solving for, 294–299 × matrix, 297–299 × matrix, 294–296 elements, 42 elimination method of algebraically solving systems of equations, 72–75 epsilon (ε), 421 equal rows or columns, zero determinants with, 206–209 equivalent matrices creating, 80 defined, 345 Euler, Leonhard (mathematician), 339 even permutations, 183 examples eigenvectors, 290 linear transformation, 148–150 Excel, 238 •F• Genetics For Dummies (Robinson), 319 genetics matrix, 318–319 geometric mean defined, 345 described, 34–35 Gram-Schmidt orthogonalization process, 281 graphing calculators Markov chain, determining results of, 334–335 for matrices, 236–237, 329–334 mode, changing, 337 place value, adjusting for a particular, 337 round function, 337 for row operations, 331–333 systems of equations, solving, 328–329, 336 graphing linear combinations of vectors, 95 graphing systems of two or three equations, 67–70 graphing two lines for two equations, 68–69 overview, 67–68 three planes meeting, 69–70 Greek letters ∆ (delta), 340 ε (epsilon), 341 λ (lambda), 341–342 μ (mu), 341 ω (omega), 342 ΦΒΚ (Phi Beta Kappa), 342 π (pi), 339–340 ρ (rho), 340 Σ (sigma), 340 θ (theta), 340 factorial operation, 176 format for polynomial, 143 free variable, 345 •H• •G• game matrix, 322–325 Gauss, Carl Friedrich (mathematician), 114 Gaussian elimination, 114 general vector equation for lines, 274 for planes, 275 Hamilton, William Rowan (mathematician), 247 heat distribution matrix, 320–321 Heron’s formula, 193 homogeneous systems of equations defined, 346 nontrivial solution, 124–128 overview, 123–124 trivial solution, 124–127 Index •I• identity, 346 identity matrix additive, 49 defined, 346 determinant of, 203–204 multiplicative, 50 overview, 49 identity transformations additive identity transformation, 159–160 multiplicative identity transformation, 160–161 image, 346 impossible statements algebraically solving systems of linear equations, 234 Cramer’s rule for solving systems of linear equations, 235–236 described, 234 row operations used for solving systems of linear equations, 234–235 inconsistent system of equations defined, 346 overview, 71–72 independent equations, 346 index, 42 index of element in matrix, 346 infinite solutions algebraically solving systems of linear equations, 235 Ax = b (matrix equation), 115–118 Cramer’s rule for solving systems of linear equations, 235–236 row operations used for solving systems of linear equations, 235 solutions for systems of equations, 67 inner product, 24, 35–37, 345 input-output models, 58 inverse matrices defined, 346 determinant of, 203–204 finding, 224–225, 228–230 overview, 58 row reduction, 59, 59–64 × 2, 59–60 used for solving systems of equations, 77–78 inversion (of a permutation) defined, 346 described, 181–182 inversions, 181–183 invertible matrix defined, 346 overview, 57–58 •K• kernel of linear transformation defined, 346 overview, 169–170 •L• lambda (λ), 341–342 least squares method, 144 Legendre, Adrien-Marie (mathematician), 144 Legendre polynomials, 144 length of vector See magnitude of vector Leontief, Wassily Wassilyovitch (inputoutput models), 58 Leontief production model, 321 linear combination defined, 346 determining whether you can produce a specific vector, 89–92 graphing, 95 overview, 87 patterns in, 93–94 scalars, 87 using orthogonal basis to write, 282–283 writing vectors as sums of other vectors, 87–88 linear dependence overview, 129 set with multiple vectors, 133–134 set with two vectors, 133 set with zero vector, 134–135 testing for, 129–131 355 356 Linear Algebra For Dummies linear equation defined, 346 described, 10, 87 linear independence characteristics of, 132–135 defined, 347 determining basis by, 141 overview, 128–129 reducing number of vectors in set with, 135 set with multiple vectors, 133–134 set with one vector, 133 set with two vectors, 133 set with unit vectors, 135 testing for, 129–132 linear operator, 148 linear programming, 76 linear transformation associative property, 154–155, 157–158 commutative property, 155–156 contractions, 168–169 defined, 347 determining when a transformation is a, 151–154 dilations, 168–169 distributive property, 161 examples, 148–150 identity transformations, 159–161 kernel (null space), 169–170 matrix for, 161–163 overview, 147–148 properties, 148 range, 170–172 reflections, 165–167 requirements, 150–151 rotations, 164–165 scalar multiplication, 150–151, 158 translations, 167–168 vector addition, 150–151 lines forming vector spaces, 274–275 general vector equation for, 274 slope-intercept form of equation of, 274 listing permutations overview, 177 with table, 177–179 with tree, 179–180 lower triangular matrix creating, 217–221 described, 51 •M• magnitude of vector defined, 347 overview, 22, 29–30 scalar multiplication, adjusting magnitude for, 30–32 triangle inequality, 32–35 main diagonal of matrix, 347 manipulated matrices adding the multiple of a row or column to another row or column, 212 multiplying a row or column by a scalar, 209–211 markers, 13 Markov chain, determining results of, 334–335 Mathematica, 238 mathematicians Cauchy, Augustin Louis, 36 Cramer, Gabriel, 233 Dantzig, George Bernard, 76 Euler, Leonhard, 339 Gauss, Carl Friedrich, 114 Hamilton, William Rowan, 247 Legendre, Adrien-Marie, 144 Vandermonde, Alexandre-Theophile, 217 matrices adjoint, 225–228 advantages of using, 13–14 basis, 142 coefficient, 77 cofactor, 226–228 column, 20, 43 column space, 265–269 commutativity, 54 computers used for calculating, 238 constant, 77 defined, 347 diagonal, 51, 304–307 dimensions of, 43 dividing, 59–60 graphing calculators for, 236–237, 329–334 Index identity, 49–50 inverse, 58, 59–64, 224–225, 228–230, 301–302 invertible, 57–58 for linear transformation, 161–163 lower triangular, 51 naming, 42 non-singular, 51 notation for, 42–43 null space, 270–271 overview, 12–14, 41 real-world applications, 312–322 row, 43 scalar multiplication of, 45 similar, 305 singular, 51 skew-symmetric, 263–264 square, 50, 51 subtraction of, 43–44 transition, 285–288, 334–335 transposing, 55–56 triangular, 51, 213 upper triangular, 51 zero, 49, 56–57 matrix addition associativity, 53 commutativity, 52–53 distributive rule, 55 overview, 43–44 matrix equation (Ax = b) infinite solutions, 115–118 multiple solutions, 112–120 no solution, 120–122 overview, 105, 110 single solution, 110–112 specialized solutions, 118–120 specific vector, solving for the solutions of a, 112–115 systems of equations and, 108–109 matrix multiplication associativity, 54 commutativity, 48, 53–54 distributive rule, 55 mixed associative rule, 54 order for, 46 overview, 45–48 requirements for, 46 two matrices, 46–48 matrix permutations, 180 matrix products, determinants of, 221–222 matrix transpose, eigenvalues of, 300 matrix-vector multiplication overview, 106–107 properties of, 108 as sum of scalar multiplications, 107–108 systems of equations and, 108–109 mean value property, 320 message encryption matrix, 315–317 methods of solving systems of linear equations, 231 minor determinants, 189–190 minor of matrix, 347 mixed associative rule, 54 Möbius function, 341 mode for graphing calculators, changing, 337 mu (µ), 341 multiple solutions for Ax = b (matrix equation), 112–120 multiplication of matrices associativity, 54 commutative, special cases in which matrix multiplication is, 48 commutativity, 53–54 distributive rule, 55 mixed associative rule, 54 order for, 46 requirements for, 46 two matrices, 46–48 multiplication symbol (×), 15 multiplicative identity matrices, 50 multiplicative identity transformation, 160–161 multiplicative inverse, 52 multiplying a row or column by a scalar, 209–211 •N• naming matrices, 42 narrowing span, 97–98 natural basis of vectors defined, 347 overview, 136–137 negative scalar multiplication, 26–27 non-collinear points, 275 non-invertible matrix, 347 357 358 Linear Algebra For Dummies non-singular matrix defined, 347 described, 51 non-spanning sets R2, 103 R3, 103–104 nonstandard position vectors, 21–23 non-trivial solution defined, 347 described, 291 determining, 127–128 norm See magnitude notation for matrices, 42–43 null space defined, 347 described, 270–271 nutrition matrix, 311–312 •O• odd permutations, 183 omega (ω), 342 order for multiplication of matrices, 46 ordered pair defined, 347 described, 73 orthogonal basis creating, 281–282 linear combination, using orthogonal basis to write the, 282–283 orthonormal basis, changing to, 283–284 orthogonal set, 280 orthogonal vectors, 37–38, 279, 347 orthogonality determining, 38 overview, 37–38 orthonormal, 428 orthonormal basis, 283–284 •P• parallelepiped defined, 348 finding volume of, 198–199 parallelogram defined, 348 described, 195 origin, finding the area when none of the vertices is at the, 196–198 origin, finding the area when one of the vertices is at the, 195–196 translation of, 197 parameter, 67, 124–125, 348 parametric equation, 348 patterns in linear combinations of vectors finding a vector set for a target vector, 93 generalizing a pattern and writing a vector set, 94 permutations defined, 348 even, 183 finding number of possible, 176–177 inversions, 181–183 listing, 177–180 matrix, 180 odd, 183 overview, 175–176 perpendicular, 348 Phi Beta Kappa (ΦΒΚ), 342 pi (π), 339–340 place value, in graphing calculators adjusting for a particular, 337 placeholders, 13 planes forming vector spaces, 275–276 general vector equation for, 275 Poisson distribution, 342 polyhedron, 348 polynomial basis, 142–144 defined, 348 described, 143 format, 143 in spanning set, 262–263 population migration matrix, 318 powers of square matrices, eigenvalues of, 303–304 predator/prey matrix, 314 properties linear transformation, 148 of matrix-vector multiplication, 108 vector space, 247–254 Index •Q• quaternions, 247 •R• R2 described, 101 non-spanning sets, 103 sets that span, 101–102 R3 described, 101 non-spanning sets, 103–104 sets that span, 102–103 range of a function f, 348 range of linear transformation defined, 348 overview, 170–172 ray defined, 348 vectors represented by, 19–21 real number, 348 real-world applications for matrices economic matrix, 321–322 game matrix, 322–325 genetics matrix, 318–319 heat distribution matrix, 320–321 message encryption matrix, 315–317 nutrition matrix, 311–312 population migration matrix, 318 predator/prey matrix, 314 spotted owl matrix, 317–318 traffic flow matrix, 312–314 reciprocal defined, 348 described, 301 reduced row echelon form, 79–82, 333 reducing number of vectors in set with linear independence, 135 reflection defined, 348 overview, 165–167 requirements linear transformation, 150–151 multiplication of matrices, 46 subspace, 256 reverse diagonal, 349 rho (ρ), 340 right angle, 349 Robinson, Tara Rodden (Genetics For Dummies), 319 rotation defined, 349 eigenvectors, 291–292 linear transformation, 164–165 round function on graphing calculators, 337 row echelon form, 79–82, 333 row matrix defined, 349 described, 43 row operations adding the multiple of one row to another, 332 adding two rows together, 331–332 echelon form, creating, 333 impossible statements, 234–235 infinite solutions, 235 interchanging rows, 204–206 multiplying a row by a scalar, 332–333 overview, 60–62, 331 switching rows, 331 row reduction, 59, 59–64 rules building subsets with, 257–258 manufacturing a matrix to replace a, 162–163 •S• scalar, 17, 24, 45, 87, 349 scalar multiplication adjusting magnitude for, 30–32 associative property, 251 closure requirement for, 245–247 commutativity, 54, 108 distributive property, 108, 251–252 linear transformation, 150–151, 158 matrix-vector multiplication as sum of scalar multiplications, 107–108 overview, 24–27, 242–245 transformation composition, 158 359 360 Linear Algebra For Dummies semiperimeter defined, 349 described, 193 set with multiple vectors, 133–134 set with one vector, 133 set with two vectors, 133 set with unit vectors, 135 set with zero vector, 134–135 sets that span R2, 101–102 R3, 102–103 sigma (∑), 420 similar matrices, 305 singular matrix defined, 349 described, 51 size of vectors, 20 skew-symmetric matrix defined, 349 overview, 263–264 slope-intercept form of a linear equation, 71–72, 274 solution of system linear equations, 349 solutions infinite, 67, 115–118, 235–236 multiple solutions, 112–120 overview, 65–66 single, 66–67, 80–82, 110–112 specialized solutions, 118–120 solutions for Ax = b (matrix equation) infinite solutions, 115–118 multiple solutions, 112–120 no solution, 120–122 overview, 110 single solution, 110–112 specialized solutions, 118–120 specific vector, solving for the solutions of a, 112–115 span broadening, 96–97 defined, 349 determining basis by, 138–141 determining whether a vector belongs, 98–101 narrowing, 97–98 overview, 95–96 spanning set bases created from, 276–279 defined, 349 finding, 261–262 overview, 261 polynomials in, 262–263 skew-symmetric matrix, 263–264 spotted owl matrix, 317–318 square matrix defined, 349 described, 50, 51 standard basis of matrices defined, 349 overview, 136–137 standard deviation, 340 standard position of a vector defined, 349 overview, 20–21 subset described, 255 determining if a set is a, 258–259 determining if you have a, 256–259 rules, building subsets with, 257–258 of vector space, 259–261 subspace defined, 349 described, 255 requirements for, 256 subtraction of matrices, 43–44 of vectors, 27, 29 summation operator, 340 superscript T, 37 superset, 255 systems of equations Ax = b (matrix equation), 108–109 consistent, 66–67 graphing calculators used for solving, 328–329, 336 inconsistent system, 71–72 matrix-vector multiplication, 108–109 overview, 10, 65 solutions for, 65–70 solving algebraically, 72–75 solving with matrices, 76–82 solving with multiple solutions, 82–83 solving with single solution, 80–82 Index •T• testing for linear dependence, 129–131 for linear independence, 129–132 theta (θ), 340–341 three planes meeting, graphing systems of two or three equations with, 69–70 × matrix determinants, computing, 187–189 eigenvalues, solving for, 297–299 eigenvectors, solving for, 297–299 three-space vectors, 23 traffic flow matrix, 312–314 transformation, 148, 429 transformation composition associative property, 157–158 overview, 156–157 scalar multiplication, 158 transition matrix, 285–288, 334–335, 350 translation defined, 350 linear transformation, 167–168 of parallelogram, 197 transpose of matrix, 350 transposing determinants, 202–203 matrices, 55–56 vectors, 37 triangle inequality in averages, 34–35 magnitude of vectors, 32–35 overview, 32–33 triangular matrices creating, 214–216 defined, 350 described, 51, 213 determinants, 213–214 eigenvalues of, 302–303 lower triangluar matrix, creating, 217–221 upper triangluar matrix, creating, 217–221 trivial solution defined, 350 determining, 126–127 nontrivial solutions compared, 124–126 overview, 124 two equations system, solving, 73–74 × matrix determinants, computing, 186–187 eigenvalues, solving for, 294–296 eigenvectors, solving for, 294–296 inverse matrices, 59–60 two-space vectors angle between, finding, 39–40 described, 20–23 •U• unit vector, 350 upper triangular matrix creating, 217–221 described, 51 •V• Vandermonde, Alexandre-Theophile (mathematician), 217 Vandermonde matrix, 217 variable, 350 variable vectors, 124 vector addition additive inverse, 253 associative property, 250–251 closure requirement for, 245–247 commutative property, 248–250 linear transformation, 150–151 overview, 27–29, 242–244 zero vector, 28–29, 253 vector in two-space, 350 vector space additive inverse, 253 closure requirement for, 245–247 defined, 242, 350 lines forming, 274–275 overview, 14–15, 241–243 planes forming, 275–276 properties, 247–254 spanning sets, 261–264, 276–279 361 362 Linear Algebra For Dummies vector space (continued) subset of, 259–261 zero vector, 253 vectors contracting, 25 on the coordinate plane, 20–23 creating, 20 defined, 430 determining whether a vector belongs to a span, 98–101 determining whether you can produce a specific vector, 89–92 dilating, 25 inner product, 35–37 length of, 22 magnitude, 22, 29–35 negative scalar multiplication, 26–27 nonstandard position, 21–23 orthogonal, 37–38 orthogonal set, 280 overview, 13, 19–20 rays, vectors represented by, 19 scalar multiplication, 24–27 size of, 20 standard position, 20–21 subtraction of, 27, 29 three-space, 23 transposing, 37 two-space, 20–23 variable, 124 zero, 26, 124 •W• writing vectors as sums of other vectors, 87–88 •Z• zero determinants with equal rows or columns, 206–209 finding a row or column of zeros, 206 overview, 206 zero matrix defined, 350 described, 49, 56–57 zero vector addition of, 28–29 described, 26, 124 vector addition, 253 vector spaces, 253 BUSINESS, CAREERS & PERSONAL FINANCE Accounting For Dummies, 4th Edition* E-Mail Marketing For Dummies Six Sigma For Dummies 978-0-470-24600-9 978-0-470-19087-6 978-0-7645-6798-8 Bookkeeping Workbook For Dummies † Job Interviews For Dummies, 3rd Edition*† 978-0-470-16983-4 978-0-470-17748-8 Small Business Kit For Dummies, 2nd Edition*† Commodities For Dummies Personal Finance Workbook For Dummies*† 978-0-7645-5984-6 978-0-470-04928-0 978-0-470-09933-9 Telephone Sales For Dummies Doing Business in China For Dummies Real Estate License Exams For Dummies 978-0-470-16836-3 978-0-470-04929-7 978-0-7645-7623-2 BUSINESS PRODUCTIVITY & MICROSOFT OFFICE Access 2007 For Dummies PowerPoint 2007 For Dummies Quicken 2008 For Dummies 978-0-470-03649-5 978-0-470-04059-1 978-0-470-17473-9 Excel 2007 For Dummies Project 2007 For Dummies 978-0-470-03737-9 978-0-470-03651-8 Salesforce.com For Dummies, 2nd Edition Office 2007 For Dummies QuickBooks 2008 For Dummies 978-0-470-04893-1 978-0-470-00923-9 978-0-470-18470-7 Word 2007 For Dummies 978-0-470-03658-7 Outlook 2007 For Dummies 978-0-470-03830-7 EDUCATION, HISTORY, REFERENCE & TEST PREPARATION African American History For Dummies ASVAB For Dummies, 2nd Edition Geometry Workbook For Dummies 978-0-7645-5469-8 978-0-470-10671-6 978-0-471-79940-5 Algebra For Dummies British Military History For Dummies The SAT I For Dummies, 6th Edition 978-0-7645-5325-7 978-0-470-03213-8 978-0-7645-7193-0 Algebra Workbook For Dummies Calculus For Dummies Series Exam For Dummies 978-0-7645-8467-1 978-0-7645-2498-1 978-0-470-09932-2 Art History For Dummies Canadian History For Dummies, 2nd Edition World History For Dummies 978-0-470-09910-0 978-0-470-83656-9 978-0-7645-5242-7 FOOD, GARDEN, HOBBIES & HOME Bridge For Dummies, 2nd Edition Drawing For Dummies Knitting Patterns For Dummies 978-0-471-92426-5 978-0-7645-5476-6 978-0-470-04556-5 Coin Collecting For Dummies, 2nd Edition Etiquette For Dummies, 2nd Edition Living Gluten-Free For Dummies † 978-0-470-22275-1 978-0-470-10672-3 978-0-471-77383-2 Cooking Basics For Dummies, 3rd Edition Gardening Basics For Dummies* † Painting Do-It-Yourself For Dummies 978-0-7645-7206-7 978-0-470-03749-2 978-0-470-17533-0 HEALTH, SELF HELP, PARENTING & PETS Anger Management For Dummies Horseback Riding For Dummies Puppies For Dummies, 2nd Edition 978-0-470-03715-7 978-0-470-09719-9 978-0-470-03717-1 Anxiety & Depression Workbook For Dummies Infertility For Dummies † Thyroid For Dummies, 2nd Edition † 978-0-470-11518-3 978-0-471-78755-6 978-0-7645-9793-0 Type Diabetes For Dummies* † Dieting For Dummies, 2nd Edition Meditation For Dummies with CD-ROM, 2nd Edition 978-0-7645-4149-0 978-0-471-77774-8 Dog Training For Dummies, 2nd Edition Post-Traumatic Stress Disorder For Dummies 978-0-7645-8418-3 978-0-470-04922-8 978-0-470-17811-9 * Separate Canadian edition also available † Separate U.K edition also available Available wherever books are sold For more information or to order direct: U.S customers visit www.dummies.com or call 1-877-762-2974 U.K customers visit www.wileyeurope.com or call (0) 1243 843291 Canadian customers visit www.wiley.ca or call 1-800-567-4797 INTERNET & DIGITAL MEDIA AdWords For Dummies eBay Business All-in-One Desk Reference For Dummies iPod & iTunes For Dummies, 5th Edition 978-0-470-15252-2 Blogging For Dummies, 2nd Edition 978-0-7645-8438-1 MySpace For Dummies 978-0-470-23017-6 eBay For Dummies, 5th Edition* 978-0-470-09529-4 Digital Photography All-in-One Desk Reference For Dummies, 3rd Edition 978-0-470-04529-9 Podcasting For Dummies eBay Listings That Sell For Dummies 978-0-471-74898-4 978-0-470-03743-0 978-0-471-78912-3 Digital Photography For Dummies, 5th Edition Facebook For Dummies Search Engine Optimization For Dummies, 2nd Edition 978-0-7645-9802-9 978-0-470-26273-3 978-0-471-97998-2 Digital SLR Cameras & Photography For Dummies, 2nd Edition The Internet For Dummies, 11th Edition Second Life For Dummies 978-0-470-12174-0 978-0-470-18025-9 978-0-470-14927-0 Investing Online For Dummies, 5th Edition Starting an eBay Business For Dummies, 3rd Edition† 978-0-7645-8456-5 978-0-470-17474-6 978-0-470-14924-9 GRAPHICS, DESIGN & WEB DEVELOPMENT Adobe Creative Suite Design Premium All-in-One Desk Reference For Dummies Creating Web Pages For Dummies, 8th Edition Photoshop CS3 For Dummies 978-0-470-11724-8 978-0-470-08030-6 Photoshop Elements For Dummies Adobe Web Suite CS3 All-in-One Desk Reference For Dummies Dreamweaver CS3 For Dummies 978-0-470-09810-3 978-0-470-11490-2 978-0-470-12099-6 SolidWorks For Dummies Flash CS3 For Dummies 978-0-7645-9555-4 AutoCAD 2008 For Dummies 978-0-470-12100-9 978-0-470-11650-0 Visio 2007 For Dummies Google SketchUp For Dummies 978-0-470-08983-5 Building a Web Site For Dummies, 3rd Edition 978-0-470-13744-4 Web Design For Dummies, 2nd Edition 978-0-470-14928-7 InDesign CS3 For Dummies 978-0-471-78117-2 978-0-470-11865-8 Web Sites Do-It-Yourself For Dummies Photoshop CS3 All-in-One Desk Reference For Dummies 978-0-470-16903-2 978-0-470-11195-6 978-0-470-17443-2 Creating Web Pages All-in-One Desk Reference For Dummies, 3rd Edition 978-0-470-09629-1 978-0-470-11193-2 Web Stores Do-It-Yourself For Dummies LANGUAGES, RELIGION & SPIRITUALITY Arabic For Dummies 978-0-471-77270-5 Chinese For Dummies, Audio Set 978-0-470-12766-7 French For Dummies 978-0-7645-5193-2 German For Dummies 978-0-7645-5195-6 Hebrew For Dummies 978-0-7645-5489-6 Ingles Para Dummies 978-0-7645-5427-8 Italian For Dummies, Audio Set 978-0-470-09586-7 Italian Verbs For Dummies 978-0-471-77389-4 Japanese For Dummies 978-0-7645-5429-2 Latin For Dummies 978-0-7645-5431-5 Portuguese For Dummies 978-0-471-78738-9 Russian For Dummies 978-0-471-78001-4 Spanish Phrases For Dummies 978-0-7645-7204-3 Spanish For Dummies 978-0-7645-5194-9 Spanish For Dummies, Audio Set 978-0-470-09585-0 The Bible For Dummies 978-0-7645-5296-0 Catholicism For Dummies 978-0-7645-5391-2 The Historical Jesus For Dummies 978-0-470-16785-4 Islam For Dummies 978-0-7645-5503-9 Spirituality For Dummies, 2nd Edition 978-0-470-19142-2 NETWORKING AND PROGRAMMING ASP.NET 3.5 For Dummies Java For Dummies, 4th Edition 978-0-470-19592-5 978-0-470-08716-9 C# 2008 For Dummies 978-0-470-05620-2 978-0-470-19109-5 Microsoft® SQL Server™ 2008 All-in-One Desk Reference For Dummies Hacking For Dummies, 2nd Edition 978-0-470-17954-3 978-0-470-09941-4 978-0-470-05235-8 Networking All-in-One Desk Reference For Dummies, 2nd Edition Wireless Home Networking For Dummies, 2nd Edition 978-0-7645-9939-2 978-0-471-74940-0 Home Networking For Dummies, 4th Edition 978-0-470-11806-1 Networking For Dummies, 8th Edition SharePoint 2007 For Dummies OPERATING SYSTEMS & COMPUTER BASICS iMac For Dummies, 5th Edition Mac OS X Leopard For Dummies 978-0-7645-8458-9 978-0-470-05433-8 Windows Vista All-in-One Desk Reference For Dummies Laptops For Dummies, 2nd Edition Macs For Dummies, 9th Edition 978-0-471-74941-7 978-0-470-05432-1 978-0-470-04849-8 Windows Vista For Dummies Linux For Dummies, 8th Edition PCs For Dummies, 11th Edition 978-0-471-75421-3 978-0-470-11649-4 978-0-470-13728-4 Windows Vista Security For Dummies MacBook For Dummies Windows® Home Server For Dummies 978-0-470-11805-4 978-0-470-04859-7 978-0-470-18592-6 Mac OS X Leopard All-in-One Desk Reference For Dummies Windows Server 2008 For Dummies 978-0-470-18043-3 978-0-470-05434-5 SPORTS, FITNESS & MUSIC Coaching Hockey For Dummies GarageBand For Dummies 978-0-470-83685-9 978-0-7645-7323-1 iPod & iTunes For Dummies, 5th Edition Coaching Soccer For Dummies Golf For Dummies, 3rd Edition 978-0-470-17474-6 978-0-471-77381-8 978-0-471-76871-5 Music Theory For Dummies Fitness For Dummies, 3rd Edition Guitar For Dummies, 2nd Edition 978-0-7645-7838-0 978-0-7645-7851-9 978-0-7645-9904-0 Stretching For Dummies Football For Dummies, 3rd Edition Home Recording For Musicians For Dummies, 2nd Edition 978-0-470-06741-3 978-0-470-12536-6 978-0-7645-8884-6 Get smart @ dummies.comđ Find a full list of Dummies titles • Look into loads of FREE on-site articles • Sign up for FREE eTips e-mailed to you weekly • See what other products carry the Dummies name • Shop directly from the Dummies bookstore • Enter to win new prizes every month! * Separate Canadian edition also available † Separate U.K edition also available Available wherever books are sold For more information or to order direct: U.S customers visit www.dummies.com or call 1-877-762-2974 U.K customers visit www.wileyeurope.com or call (0) 1243 843291 Canadian customers visit www.wiley.ca or call 1-800-567-4797 s e i m m u D h t i w e r o M o D ! s U f o t s e R e h t r o f s t c Produ DVDs • Music • Games • DIY Consumer Electronics • Software • Crafts Hobbies • Cookware • and more! Check out the Dummies Product Shop at www.dummies.com for more information! Mathematics/Algebra Your hands-on guide to real-world applications of linear algebra Does linear algebra leave you feeling lost? No worries — this easy-to-follow guide explains the how and the why of solving linear algebra problems in plain English From matrices to vector spaces to linear transformations, you’ll understand the key concepts and see how they relate to everything from genetics to nutrition to spotted owl extinction • Line up the basics — discover several different approaches to organizing numbers and equations, and solve systems of equations algebraically or with matrices • Relate vectors and linear transformations — link vectors and matrices with linear combinations and seek solutions of homogeneous systems Open the book and find: • Theoretical and practical ways of solving linear algebra problems • Definitions of terms throughout and in the glossary • New ways of looking at operations • How linear algebra ties together vectors, matrices, determinants, and linear transformations • Ten common mathematical representations of Greek letters • Real-world applications of matrices and determinants • Evaluate determinants — see how to perform the determinant function on different sizes of matrices and take advantage of Cramer’s rule • Hone your skills with vector spaces — determine the properties of vector spaces and their subspaces and see linear transformation in action • Tackle eigenvalues and eigenvectors — define and solve for eigenvalues and eigenvectors and understand how they interact with specific matrices Go to dummies.com® for more! $19.99 US / $23.99 CN / £14.99 UK Mary Jane Sterling is the author of numerous For Dummies books She is a lecturer at Bradley University in Peoria, Illinois, where she has taught courses in algebra, calculus, and other mathematics topics for almost 30 years ISBN 978-0-470-43090-3 ... five other For Dummies titles (all published by Wiley): Algebra For Dummies, Algebra II For Dummies, Trigonometry For Dummies, Math Word Problems For Dummies, and Business Math For Dummies Mary... Linear Algebra FOR DUMmIES ‰ Linear Algebra FOR DUMmIES ‰ by Mary Jane Sterling Linear Algebra For Dummies Published by Wiley Publishing, Inc 111... .147 Formulating Linear Transformations 147 Delineating linear transformation lingo 148 Recognizing when a transformation is a linear transformation 151 Proposing Properties of Linear