Linear Algebra

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$MATLAB C++ Math Library Reference$

estriction, A-9 right inverse, 231 structure preserving, 161, 165 seehomomorphism, 176 two-sided inverse, 231 well-defined, A-8 zero, 177 Fundamental Theorem of Linear Algebra, 268 Gauss’ method, 2 accuracy, 67–71 back-substitution, 5 elementary operations, 4 Gauss-Jordan, 46 Gauss-Jordan, 46 generalized nullspace, 367 generalized rangespace, 367 Geometry of Linear Maps, 274–279 Gram-Schmidt process, 255–260 homogeneous coordinate vector, 340 homogeneous coordinates, 291 homogeneous equation, 21 homomorphism, 176 composition, 215 matrix representing, 194–204 nonsingular, 190, 207 nullity, 188 nullspace, 188 rangespace, 184 rank, 206 zero, 177 ideal line, 342 ideal point, 342 identity function, A-8 matrix, 224 ill-conditioned, 68 implication, A-2 improper subspace, 92 incidence matrix, 228 index of nilpotency, 370 induced map, 275 induction, 23, A-5 inner product, 39 Input-Output Analysis, 63–66 internal direct sum, 135 invariant subspace, 377 invariant subspace definition, 389 inverse, 231, A-9 additive, 80 exists, 231 left, 231, A-9 matrix, 329 right, 231, A-9 two-sided, A-9 inverse function, 231 inverse image, 185 inversion, 313 isometry, 286 isomorphism, 159–175 characterized by dimension, 172 definition, 161 of a space with itself, 163 Jordan block, 388 Jordan form, 379–398 represents similarity classes, 392 kernel, 188 Kirchhoff’s Laws, 73 Klein, F., 286 Laplace expansion, 326–330 computes determinant, 327 leading variable, 5 least squares, 269–273 length, 39 Leontief, W., 63 line best fit, 269 in projective plane, 341 line at infinity, 342 line of best fit, 269–273 linear transpose operation, 131 linear combination, 52 Linear Combination Lemma, 52 linear equation, 2 coefficients, 2 constant, 2 homogeneous, 21 inconsistent systems, 269 satisfied by a vector, 15 solution of, 2
Gauss’ method, 3 Gauss-Jordan, 46 solutions of Cramer’s rule, 332 system of, 2 linear map dilation, 275 reflection, 289 rotation, 274, 288 seehomomorphism, 176 skew, 276 trace, 397 linear recurrence, 406 linear recurrences, 405–412 linear relationship, 103 linear surface, 36 linear transformation seetransformation, 180 linearly dependent, 103 linearly independent, 103 LINPACK, 61 map distance-preserving, 286 extended linearly, 173 induced, 275 self composition, 365 Maple, 61 Markov chains, 280–285 Markov matrix, 284 Mathematica, 61 mathematical induction, 23 MATLAB, 61 matrix, 13 adjoint, 328 antisymmetric, 139 augmented, 14 block, 246, 311 change of basis, 238 characteristic polynomial, 360 cofactor, 327 column, 13 column space, 126 conditioning number, 70 determinant, 294, 299 diagonal, 209, 225 diagonalizable, 354 diagonalized, 244 elementary reduction, 226, 275 entry, 13 equivalent, 244 identity,220,224 incidence, 228 induced map, 275 inverse, 329 main diagonal, 224 Markov, 229, 284 matrix-vector product, 197 minimal polynomial, 220, 380 minor, 327 multiplication, 215 nilpotent, 370 nonsingular, 27, 207 orthogonal, 288 orthonormal, 286–291 permutation, 225 rank, 206 representation, 196 row, 13 row equivalence, 50 row rank, 124 row space, 124 scalar multiple, 212 similar, 324 similarity, 351 singular, 27 skew-symmetric, 311 submatrix, 303 sum, 212 symmetric, 118, 139, 213, 220, 228, 268 trace, 213, 229, 397 transpose, 19, 126, 213 triangular, 204, 229, 330 unit, 222 Vandermonde, 311 matrix equivalence, 242–249 canonical form, 245 definition, 244 matrix:form, 55 mean arithmetic, 44 geometric, 44 method of powers, 399–402 minimal polynomial, 220, 380 minor, 327 morphism, 161 multilinear, 305 multiplication matrix-matrix, 215
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Preface In most mathematics program
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dent projects for individuals or sm
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Contents 1 Linear Systems 1 1.I Sol
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5.II.1 Definition and Examples ....
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2 Chapter 1. Linear Systems To fini
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4 Chapter 1. Linear Systems Conside
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6 Chapter 1. Linear Systems the sec
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8 Chapter 1. Linear Systems Don’t
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10 Chapter 1. Linear Systems 1.25 G
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12 Chapter 1. Linear Systems can al
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14 Chapter 1. Linear Systems Matric
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16 Chapter 1. Linear Systems Scalar
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18 Chapter 1. Linear Systems Must t
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20 Chapter 1. Linear Systems 2.30 [
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22 Chapter 1. Linear Systems Studyi
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24 Chapter 1. Linear Systems the bo
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26 Chapter 1. Linear Systems shows
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28 Chapter 1. Linear Systems 3.13 E
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30 Chapter 1. Linear Systems (a) 2x
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32 Chapter 1. Linear Systems 1.II L
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34 Chapter 1. Linear Systems positi
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36 Chapter 1. Linear Systems In R3
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38 Chapter 1. Linear Systems � 1.
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40 Chapter 1. Linear Systems Notice
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42 Chapter 1. Linear Systems 2.7 De
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44 Chapter 1. Linear Systems 2.25 P
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46 Chapter 1. Linear Systems We can
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48 Chapter 1. Linear Systems each e
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54 Chapter 1. Linear Systems x1 has
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56 Chapter 1. Linear Systems First,
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58 Chapter 1. Linear Systems 2.8 Ex
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60 Chapter 1. Linear Systems (2) Ca
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62 Chapter 1. Linear Systems (b) Th
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64 Chapter 1. Linear Systems For th
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66 Chapter 1. Linear Systems (a) Fi
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68 Chapter 1. Linear Systems 4 3 2
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70 Chapter 1. Linear Systems expert
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72 Chapter 1. Linear Systems Topic:
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74 Chapter 1. Linear Systems We beg
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76 Chapter 1. Linear Systems 1 Calc
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78 Chapter 1. Linear Systems parall
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80 Chapter 2. Vector Spaces 2.I Def
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82 Chapter 2. Vector Spaces For the
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84 Chapter 2. Vector Spaces A vecto
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86 Chapter 2. Vector Spaces 1.12 Ex
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88 Chapter 2. Vector Spaces Chapter
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90 Chapter 2. Vector Spaces (d) The
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92 Chapter 2. Vector Spaces 2.2 Exa
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94 Chapter 2. Vector Spaces second.
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96 Chapter 2. Vector Spaces Proof.
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98 Chapter 2. Vector Spaces The sub
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100 Chapter 2. Vector Spaces S. Mem
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102 Chapter 2. Vector Spaces 2.II L
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104 Chapter 2. Vector Spaces 1.5 Ex
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106 Chapter 2. Vector Spaces Lookin
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108 Chapter 2. Vector Spaces Proof.
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110 Chapter 2. Vector Spaces (a) {2
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112 Chapter 2. Vector Spaces � 1.
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114 Chapter 2. Vector Spaces 1.5 De
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116 Chapter 2. Vector Spaces 1.13 D
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118 Chapter 2. Vector Spaces � 1.
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120 Chapter 2. Vector Spaces we stu
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122 Chapter 2. Vector Spaces 2.11 C
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124 Chapter 2. Vector Spaces subspa
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126 Chapter 2. Vector Spaces 3.6 De
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128 Chapter 2. Vector Spaces Anothe
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130 Chapter 2. Vector Spaces (a)
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132 Chapter 2. Vector Spaces by fin
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134 Chapter 2. Vector Spaces has a
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136 Chapter 2. Vector Spaces 4.12 E
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138 Chapter 2. Vector Spaces 4.22 I
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140 Chapter 2. Vector Spaces (c) Le
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142 Chapter 2. Vector Spaces We cou
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144 Chapter 2. Vector Spaces Then w
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146 Chapter 2. Vector Spaces (d) Pl
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148 Chapter 2. Vector Spaces howeve
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150 Chapter 2. Vector Spaces positi
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152 Chapter 2. Vector Spaces Topic:
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154 Chapter 2. Vector Spaces The cl
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156 Chapter 2. Vector Spaces Remark
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158 Chapter 2. Vector Spaces (b) Sh
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160 Chapter 3. Maps Between Spaces
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Chapter 4 Determinants In the first
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Section I. Definition 295 determine
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Section I. Definition 297 Exercises
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Section I. Definition 299 1.15 Prov
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Section I. Definition 301 That resu
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Section I. Definition 303 (b) Prove
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Section I. Definition 305 3.2 Defin
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Section I. Definition 307 Therefore
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Section I. Definition 309 read alou
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Section I. Definition 311 � 3.23
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Section I. Definition 313 could be
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Section I. Definition 315 We still
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Section I. Definition 317 (terms wi
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Section II. Geometry of Determinant
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Section III. Other Formulas 327 The
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Section III. Other Formulas 329 1.9
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Topic: Cramer’s Rule 331 Topic: C
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Topic: Cramer’s Rule 333 4 Suppos
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Topic: Speed of Calculating Determi
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Topic: Projective Geometry 337 Topi
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Topic: Projective Geometry 339 P P
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Topic: Projective Geometry 341 of t
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Topic: Projective Geometry 343 The
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Topic: Projective Geometry 345 the
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Chapter 5 Similarity While studying
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Section I. Complex Vector Spaces 34
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Section II. Similarity 351 5.II Sim
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Section II. Similarity 353 1.7 Cons
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Section II. Similarity 355 2.4 Coro
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Section II. Similarity 357 2.12 The
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Section II. Similarity 359 and the
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Section II. Similarity 361 Proof. A
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Section II. Similarity 363 3.19 Cor
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Section III. Nilpotence 365 5.III N
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Section III. Nilpotence 367 and thi
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Section III. Nilpotence 369 Proof.
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Section III. Nilpotence 371 The new
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Section III. Nilpotence 373 Now, wi
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Section III. Nilpotence 375 We can
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Section III. Nilpotence 377 � −
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Section IV. Jordan Form 379 5.IV Jo
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Page 409 and 410: Topic: Computing Eigenvalues—the
Page 411 and 412: Topic: Computing Eigenvalues—the
Page 413 and 414: Topic: Stable Populations 403 Topic
Page 415 and 416: Topic: Linear Recurrences 405 Topic
Page 417 and 418: Topic: Linear Recurrences 407 And,
Page 419 and 420: Topic: Linear Recurrences 409 diamo
Page 421 and 422: Topic: Linear Recurrences 411 Compu
Page 423 and 424: Appendix Introduction Mathematics i
Page 425 and 426: A-3 ‘if a number is divisible by
Page 427 and 428: Techniques of Proof A-5 Induction.
Page 429 and 430: A-7 The Principle of Extensionality
Page 431 and 432: A-9 So an identity map plays the sa
Page 433 and 434: A-11 Similarly, the equivalence rel
Page 435 and 436: Bibliography [Abbot] Edwin Abbott,
Page 437 and 438: [Feller] William Feller, An Introdu
Page 439 and 440: [Programmer’s Ref.] Microsoft Pro
Page 441: congruent plane figures, 286 contra
Page 445 and 446: educed echelon form, 46 reflection,
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Linear Algebra

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