Linear Algebra

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$MATLAB C++ Math Library Reference$

290 Chapter 3. Maps Between Spaces Frame 1: Frame 2: Frame 3: −.2 radians −.4 radians −.6 radians We could also make the cube appear to be coming closer to us by producing film frames of it gradually enlarging. Frame 1: Frame 2: Frame 3: 110 percent 120 percent 130 percent In practice, computer graphics incorporate many interesting techniques from linear algebra (see Exercise 4). So the analysis above of distance-preserving maps is useful as well as interesting. For instance, it shows that to include in graphics software all possible rigid motions of the plane, we need only include a few cases. It is not possible that we’ve somehow ovelooked some rigid motions. A beautiful book that explores more in this area is [Weyl]. More on groups, of transformations and otherwise, can be found in any book on Modern <strong>Algebra</strong>, for instance [Birkhoff & MacLane]. More on Klein and the Erlanger Program is in [Yaglom]. Exercises 1 Decide � if √each of these √ is an orthonormal matrix. 1/ 2 −1/ 2 (a) −1/ √ 2 −1/ √ � 2 � √ √ 1/ 3 −1/ 3 (b) −1/ √ 3 −1/ √ � 3 � √ √ √ 1/ 3 − 2/ 3 (c) − √ 2/ √ 3 −1/ √ � 3 2 Write down the formula for each of these distance-preserving maps. (a) the map that rotates π/6 radians, and then translates by �e2 (b) the map that reflects about the line y =2x (c) the map that reflects about y = −2x andtranslatesover1andup1 3 (a) The proof that a map that is distance-preserving and sends the zero vector to itself incidentally shows that such a map is one-to-one and onto (the point in the domain determined by d0, d1, and d2 corresponds to the point in the codomain determined by those three numbers). Therefore any distancepreserving map has an inverse. Show that the inverse is also distance-preserving. (b) Using the definitions given in this Topic, prove that congruence is an equivalence relation between plane figures.
Topic: Orthonormal Matrices 291 4 In practice the matrix for the distance-preserving linear transformation and the translation are often combined into one. Check that these two computations yield the same first two components. � �� a c x e + b d y f � a b 0 c d 0 �� e x f y 1 1 (These are homogeneous coordinates; see the Topic on Projective Geometry). 5 (a) Verify that the properties described in the second paragraph of this Topic as invariant under distance-preserving maps are indeed so. (b) Give two more properties that are of interest in Euclidean geometry from your experience in studying that subject that are also invariant under distancepreserving maps. (c) Give a property that is not of interest in Euclidean geometry and is not invariant under distance-preserving maps.
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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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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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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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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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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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Topic: Projective Geometry 341 of t
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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 359 and the
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Section II. Similarity 361 Proof. A
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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 377 � −
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Section IV. Jordan Form 379 5.IV Jo
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Section IV. Jordan Form 381 Using t
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Section IV. Jordan Form 383 Equate
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Section IV. Jordan Form 385 � 1.1
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Section IV. Jordan Form 387 is (x
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Section IV. Jordan Form 393 2.13 Ex
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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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