Elementary Linear Algebra

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