Linear Algebra

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$MATLAB C++ Math Library Reference$

402 Chapter 5. Similarity 39367 >v10=T10*v0 v10= 59049 118096 >norm(v10)/norm(v9) ans=2.9999 (It has been lightly edited to remove blank lines, etc. Remark: we are ignoring the power of Octave here; there are built-in functions to automatically apply quite sophisticated methods to find eigenvalues and eigenvectors. Instead, we are using just the system as a calculator.)
Topic: Stable Populations 403 Topic: Stable Populations Imagine a reserve park with animals from a species that we are trying to protect. The park doesn’t have a fence and so animals cross the boundary, both from the inside out and in the other direction. Every year, 10% of the animals from inside of the park leave, and 1% of the animals from the outside find their way in. We can ask if we can find a stable level of population for this park: is there a population that, once established, will stay constant over time, with the number of animals leaving equal to the number of animals entering? To answer that question, we must first establish the equations. Let the year n population in the park be pn and in the rest of the world be rn. pn+1 = .90pn + .01rn rn+1 = .10pn + .99rn We can set this system up as a matrix equation (see the Markov Chain topic). � � � �� pn+1 .90 .01 pn = .10 .99 rn+1 Now, “stable level” means that pn+1 = pn and rn+1 = rn, so that the matrix equation �vn+1 = T�vn becomes �v = T�v. We are therefore looking for eigenvectors for T that are associated with the eigenvalue 1. The equation (I − T )�v = �0 is � �� .10 .01 p 0 = .10 .01 r 0 which gives the eigenspace: vectors with the restriction that p = .1r. Coupled with additional information, that the total world population of this species is is p + r = 110 000, we find that the stable state is p =10, 000 and r = 100, 000. If we start with a park population of ten thousand animals, so that the rest of the world has one hundred thousand, then every year ten percent (a thousand animals) of those inside will leave the park, and every year one percent (a thousand) of those from the rest of the world will enter the park. It is stable, self-sustaining. Now imagine that we are trying to gradually build up the total world population of this species. We can try, for instance, to have the world population grow at a rate of 1% per year. In this case, we can take a “stable” state for the park’s population to be that it also grows at 1% per year. The equation �vn+1 =1.01 · �vn = T�vn leads to ((1.01 · I) − T )�v = �0, which gives this system. � �� .11 .01 p = .10 .02 r rn � � 0 0 The matrix is nonsingular, and so the only solution is p =0andr =0. Thus, there is no (usable) initial population that we can establish at the park and expect that it will grow at the same rate as the rest of the world.
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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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14 Chapter 1. Linear Systems Matric
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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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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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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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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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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 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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Chapter 5 Similarity While studying
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Section I. Complex Vector Spaces 34
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Page 409 and 410: Topic: Computing Eigenvalues—the
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Page 415 and 416: Topic: Linear Recurrences 405 Topic
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Page 421 and 422: Topic: Linear Recurrences 411 Compu
Page 423 and 424: Appendix Introduction Mathematics i
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Page 427 and 428: Techniques of Proof A-5 Induction.
Page 429 and 430: A-7 The Principle of Extensionality
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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 and 442: congruent plane figures, 286 contra
Page 443 and 444: Gauss’ method, 3 Gauss-Jordan, 46
Page 445 and 446: educed echelon form, 46 reflection,
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Linear Algebra

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