Linear Algebra

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$MATLAB C++ Math Library Reference$

3 Maps Between Spaces 159 3.I Isomorphisms ..............................159 3.I.1 Definition and Examples ......................159 3.I.2 Dimension Characterizes Isomorphism ..............169 3.II Homomorphisms ............................176 3.II.1 Definition ..............................176 3.II.2 Rangespace and Nullspace .....................184 3.III Computing Linear Maps ........................194 3.III.1 Representing Linear Maps with Matrices ............194 3.III.2 Any Matrix Represents a Linear Map ∗ ..............204 3.IV Matrix Operations ...........................211 3.IV.1 Sums and Scalar Products .....................211 3.IV.2 Matrix Multiplication .......................214 3.IV.3 Mechanics of Matrix Multiplication ................221 3.IV.4 Inverses ...............................230 3.V Change of Basis ............................238 3.V.1 Changing Representations of Vectors ...............238 3.V.2 Changing Map Representations ..................242 3.VI Projection ................................250 3.VI.1 Orthogonal Projection Into a Line ∗ ................250 3.VI.2 Gram-Schmidt Orthogonalization ∗ ................255 3.VI.3 Projection Into a Subspace ∗ ....................260 Topic: Line of Best Fit ...........................269 Topic: Geometry of Linear Maps ......................274 Topic: Markov Chains ............................280 Topic: Orthonormal Matrices ........................286 4 Determinants 293 4.I Definition ................................294 4.I.1 Exploration ∗ ............................294 4.I.2 Properties of Determinants ....................299 4.I.3 The Permutation Expansion ....................303 4.I.4 Determinants Exist ∗ ........................312 4.II Geometry of Determinants ......................319 4.II.1 Determinants as Size Functions ..................319 4.III Other Formulas .............................326 4.III.1 Laplace’s Expansion ∗ .......................326 Topic: Cramer’s Rule ............................331 Topic: Speed of Calculating Determinants .................334 Topic: Projective Geometry .........................337 5 Similarity 347 5.I Complex Vector Spaces ........................347 5.I.1 Factoring and Complex Numbers; A Review ∗ ..........348 5.I.2 Complex Representations .....................350 5.II Similarity ................................351 vi
5.II.1 Definition and Examples ......................351 5.II.2 Diagonalizability ..........................353 5.II.3 Eigenvalues and Eigenvectors ...................357 5.III Nilpotence ...............................365 5.III.1 Self-Composition ∗ .........................365 5.III.2 Strings ∗ ...............................368 5.IV Jordan Form ..............................379 5.IV.1 Polynomials of Maps and Matrices ∗ ...............379 5.IV.2 Jordan Canonical Form ∗ ......................386 Topic: Computing Eigenvalues—the Method of Powers .........399 Topic: Stable Populations ..........................403 Topic: Linear Recurrences .........................405 Appendix A-1 Introduction .................................A-1 Propositions .................................A-1 Quantifiers .................................A-3 Techniques of Proof ............................A-5 Sets, Functions, and Relations .......................A-6 ∗ Note: starred subsections are optional. vii
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56 Chapter 1. Linear Systems First,
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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 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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Chapter 5 Similarity While studying
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Section II. Similarity 361 Proof. A
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Section III. Nilpotence 369 Proof.
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Section IV. Jordan Form 379 5.IV Jo
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Section IV. Jordan Form 387 is (x
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Section IV. Jordan Form 397 � 5 4
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Topic: Computing Eigenvalues—the
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Topic: Computing Eigenvalues—the
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Topic: Stable Populations 403 Topic
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Topic: Linear Recurrences 405 Topic
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Topic: Linear Recurrences 407 And,
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Topic: Linear Recurrences 409 diamo
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Topic: Linear Recurrences 411 Compu
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Appendix Introduction Mathematics i
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A-3 ‘if a number is divisible by
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Techniques of Proof A-5 Induction.
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A-7 The Principle of Extensionality
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A-9 So an identity map plays the sa
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A-11 Similarly, the equivalence rel
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Bibliography [Abbot] Edwin Abbott,
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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congruent plane figures, 286 contra
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Gauss’ method, 3 Gauss-Jordan, 46
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educed echelon form, 46 reflection,
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Linear Algebra

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