Elementary linear algebra : students solutions manual

The cornerstone of ELEMENTARY LINEAR ALGEBRA is the authors' clear, careful, and concise presentation of material—written so that readers can fully understand how mathematics works. This program balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system. Featuring a new design that highlights the relevance of the mathematics and improves readability, the Seventh Edition also incorporates new conceptual Capstone exercises that reinforce multiple concepts in each section. Data and applications reflect current statistics and examples to engage users and demonstrate the link between theory and practice. Cover......Page 1 Title Page......Page 7 Copyright......Page 8 Contents......Page 9 Preface......Page 13 Instructor Resources......Page 16 Student Resources......Page 17 Acknowledgements......Page 18 1 Systems of Linear Equations......Page 19 LINEAR EQUATIONS IN n VARIABLES......Page 20 SOLUTIONS AND SOLUTION SETS......Page 21 SYSTEMS OF LINEAR EQUATIONS......Page 22 SOLVING A SYSTEM OF LINEAR EQUATIONS......Page 24 MATRICES......Page 31 ELEMENTARY ROW OPERATIONS......Page 32 GAUSS-JORDAN ELIMINATION......Page 37 HOMOGENEOUS SYSTEMS OF LINEAR EQUATIONS......Page 39 POLYNOMIAL CURVE FITTING......Page 43 NETWORK ANALYSIS......Page 47 Review Exercises......Page 53 Project 2 Underdetermined and Overdetermined Systems......Page 56 2 Matrices......Page 57 EQUALITY OF MATRICES......Page 58 MATRIX ADDITION, SUBTRACTION, AND SCALAR MULTIPLICATION......Page 59 MATRIX MULTIPLICATION......Page 60 SYSTEMS OF LINEAR EQUATIONS......Page 63 PARTITIONED MATRICES......Page 64 ALGEBRA OF MATRICES......Page 70 PROPERTIES OF MATRIX MULTIPLICATION......Page 72 THE TRANSPOSE OF A MATRIX......Page 75 MATRICES AND THEIR INVERSES......Page 80 PROPERTIES OF INVERSES......Page 85 SYSTEMS OF EQUATIONS......Page 88 ELEMENTARY MATRICES AND ELEMENTARY ROW OPERATIONS......Page 92 THE LU -FACTORIZATION......Page 97 STOCHASTIC MATRICES......Page 102 CRYPTOGRAPHY......Page 105 LEONTIEF INPUT-OUTPUT MODELS......Page 108 LEAST SQUARES REGRESSION ANALYSIS......Page 110 Review Exercises......Page 116 Project 2 Nilpotent Matrices......Page 120 3 Determinants......Page 121 THE DETERMINANT OF A 2 x 2 MATRIX......Page 122 THE DETERMINANT OF A SQUARE MATRIX......Page 124 TRIANGULAR MATRICES......Page 127 DETERMINANTS AND ELEMENTARY ROW OPERATIONS......Page 130 DETERMINANTS AND ELEMENTARY COLUMN OPERATIONS......Page 132 MATRICES AND ZERO DETERMINANTS......Page 133 MATRIX PRODUCTS AND SCALAR MULTIPLES......Page 138 DETERMINANTS AND THE INVERSE OF A MATRIX......Page 140 DETERMINANTS AND THE TRANSPOSE OF A MATRIX......Page 142 THE ADJOINT OF A MATRIX......Page 146 CRAMER’S RULE......Page 148 AREA, VOLUME, AND EQUATIONS OF LINES AND PLANES......Page 150 Review Exercises......Page 156 Project 2 The Cayley-Hamilton Theorem......Page 159 Cumulative Test for Chapters 1–3......Page 161 4 Vector Spaces......Page 163 VECTORS IN THE PLANE......Page 164 VECTOR OPERATIONS......Page 165 VECTORS IN R[Sup(n)]......Page 167 LINEAR COMBINATIONS OF VECTORS......Page 170 DEFINITION OF A VECTOR SPACE......Page 173 SETS THAT ARE NOT VECTOR SPACES......Page 177 SUBSPACES......Page 180 SUBSPACES OF R[Sup(n)]......Page 183 LINEAR COMBINATIONS OF VECTORS IN A VECTOR SPACE......Page 187 SPANNING SETS......Page 189 LINEAR DEPENDENCE AND LINEAR INDEPENDENCE......Page 191 BASIS FOR A VECTOR SPACE......Page 198 THE DIMENSION OF A VECTOR SPACE......Page 203 ROW SPACE, COLUMN SPACE, AND RANK OF A MATRIX......Page 207 THE NULLSPACE OF A MATRIX......Page 212 SOLUTIONS OF SYSTEMS OF LINEAR EQUATIONS......Page 215 COORDINATE REPRESENTATION IN R[Sup(n)]......Page 220 CHANGE OF BASIS IN R[Sup(n)]......Page 222 COORDINATE REPRESENTATION IN GENERAL n-DIMENSIONAL SPACES......Page 227 LINEAR DIFFERENTIAL EQUATIONS (CALCULUS)......Page 230 CONIC SECTIONS AND ROTATION......Page 233 Review Exercises......Page 239 Project 2 Direct Sum......Page 242 5 Inner Product Spaces......Page 243 VECTOR LENGTH AND UNIT VECTORS......Page 244 DISTANCE BETWEEN TWO VECTORS IN R[Sup(n)]......Page 246 DOT PRODUCT AND THE ANGLE BETWEEN TWO VECTORS......Page 247 THE DOT PRODUCT AND MATRIX MULTIPLICATION......Page 252 INNER PRODUCTS......Page 255 ORTHOGONAL PROJECTIONS IN INNER PRODUCT SPACES......Page 261 ORTHOGONAL AND ORTHONORMAL SETS......Page 266 GRAM-SCHMIDT ORTHONORMALIZATION PROCESS......Page 271 THE LEAST SQUARES PROBLEM......Page 277 ORTHOGONAL SUBSPACES......Page 278 FUNDAMENTAL SUBSPACES OF A MATRIX......Page 282 SOLVING THE LEAST SQUARES PROBLEM......Page 283 MATHEMATICAL MODELING......Page 285 THE CROSS PRODUCT OF TWO VECTORS IN R[Sup(3)]......Page 289 LEAST SQUARES APPROXIMATIONS (CALCULUS)......Page 293 FOURIER APPROXIMATIONS (CALCULUS)......Page 297 Review Exercises......Page 302 Project 1 The QR-Factorization......Page 305 Project 2 Orthogonal Matrices and Change of Basis......Page 306 Cumulative Test for Chapters 4 and 5......Page 307 6 Linear Transformations......Page 309 IMAGES AND PREIMAGES OF FUNCTIONS......Page 310 LINEAR TRANSFORMATIONS......Page 311 THE KERNEL OF A LINEAR TRANSFORMATION......Page 321 THE RANGE OF A LINEAR TRANSFORMATION......Page 324 ONE-TO-ONE AND ONTO LINEAR TRANSFORMATIONS......Page 327 ISOMORPHISMS OF VECTOR SPACES......Page 329 THE STANDARD MATRIX FOR A LINEAR TRANSFORMATION......Page 332 COMPOSITION OF LINEAR TRANSFORMATIONS......Page 335 NONSTANDARD BASES AND GENERAL VECTOR SPACES......Page 338 THE MATRIX FOR A LINEAR TRANSFORMATION......Page 342 SIMILAR MATRICES......Page 344 THE GEOMETRY OF LINEAR TRANSFORMATIONS IN R[Sup(2)]......Page 348 ROTATION IN R[Sup(3)]......Page 351 Review Exercises......Page 355 Project 2 Reflections in R[Sup(2)] (II)......Page 358 7 Eigenvalues and Eigenvectors......Page 359 THE EIGENVALUE PROBLEM......Page 360 FINDING EIGENVALUES AND EIGENVECTORS......Page 363 EIGENVALUES AND EIGENVECTORS OF LINEAR TRANSFORMATIONS......Page 367 THE DIAGONALIZATION PROBLEM......Page 371 DIAGONALIZATION AND LINEAR TRANSFORMATIONS......Page 377 SYMMETRIC MATRICES......Page 380 ORTHOGONAL MATRICES......Page 382 ORTHOGONAL DIAGONALIZATION......Page 385 POPULATION GROWTH......Page 390 SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS (CALCULUS)......Page 392 QUADRATIC FORMS......Page 394 Review Exercises......Page 403 Project 2 The Fibonacci Sequence......Page 406 Cumulative Test for Chapters 6 and 7......Page 407 Appendix: Mathematical Induction and Other Forms of Proofs......Page 409 Index......Page 447