Linear Algebra

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398 Chapter Five. Similarity1968339367>v10=T10*v0v10=59049118096>norm(v10)/norm(v9)ans=2.9999Remark: we are ignoring the power of Octave here; there are built-in functionsto automatically apply quite sophisticated methods to find eigenvalues andeigenvectors. Instead, we are using just the system as a calculator.
Topic: Stable Populations 399Topic: Stable PopulationsImagine 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 fromthe inside out and in the other direction. Every year, 10% of the animals frominside of the park leave, and 1% of the animals from the outside find their wayin. We can ask if we can find a stable level of population for this park: is there apopulation that, once established, will stay constant over time, with the numberof animals leaving equal to the number of animals entering?To answer that question, we must first establish the equations. Let the yearn population in the park be p n and in the rest of the world be r n .p n+1 = .90p n + .01r nr n+1 = .10p n + .99r nWe can set this system up as a matrix equation (see the Markov Chain topic).( )pn+1=r n+1( ) ( ).90 .01 pn.10 .99 r nNow, “stable level” means that p n+1 = p n and r n+1 = r n , so that the matrixequation ⃗v n+1 = T⃗v n becomes ⃗v = T⃗v. We are therefore looking for eigenvectorsfor T that are associated with the eigenvalue 1. The equation (I − T )⃗v = ⃗0 is( ( ) ( .10 p 0=.10 .01)r 0)which gives the eigenspace: vectors with the restriction that p = .1r. Coupledwith additional information, that the total world population of this species is isp + 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 ofthe world has one hundred thousand, then every year ten percent (a thousandanimals) of those inside will leave the park, and every year one percent (athousand) 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 populationof this species. We can try, for instance, to have the world populationgrow at a rate of 1% per year. In this case, we can take a “stable” state forthe park’s population to be that it also grows at 1% per year. The equation⃗v n+1 = 1.01 · ⃗v n = T⃗v n leads to ((1.01 · I) − T )⃗v = ⃗0, which gives this system.( ( ) ( .11 .01 p 0=.10 .02)r 0)The matrix is nonsingular, and so the only solution is p = 0 and r = 0. Thus,there is no (usable) initial population that we can establish at the park andexpect that it will grow at the same rate as the rest of the world.
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Linear AlgebraJim Hefferon( 13)( 21
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PrefaceThis book helps students to
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the material, but it’s only that
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Chapter Three: Maps Between Spaces
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Chapter OneLinear SystemsISolving L
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Section I. Solving Linear Systems 3
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Topic: Computer Algebra Systems 631
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Topic: Input-Output Analysis 67(c)
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Topic: Accuracy of Computations 69(
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Topic: Analyzing Networks 73differe
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Topic: Analyzing Networks 75Kirchof
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80 Chapter Two. Vector SpacesIDefin
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82 Chapter Two. Vector SpacesThe ni
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84 Chapter Two. Vector SpacesWarnin
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86 Chapter Two. Vector Spaces1.13 E
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88 Chapter Two. Vector SpacesExerci
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92 Chapter Two. Vector Spaces2.3 Ex
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94 Chapter Two. Vector Spacesnonemp
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98 Chapter Two. Vector Spaceš 2.2
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100 Chapter Two. Vector Spaces⃗v
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104 Chapter Two. Vector Spaces1.9 R
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106 Chapter Two. Vector SpacesResta
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108 Chapter Two. Vector SpacesIn de
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110 Chapter Two. Vector Spaces1.35
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112 Chapter Two. Vector SpacesIIIBa
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120 Chapter Two. Vector Spacesthat
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126 Chapter Two. Vector SpacesNow t
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128 Chapter Two. Vector SpacesSo ou
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134 Chapter Two. Vector SpacesProof
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138 Chapter Two. Vector Spaces(b) A
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140 Chapter Two. Vector SpacesTopic
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144 Chapter Two. Vector Spaces(To s
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148 Chapter Two. Vector Spacesdimen
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288 Chapter Four. DeterminantsIDefi
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292 Chapter Four. Determinantš 1.
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294 Chapter Four. DeterminantsProof
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296 Chapter Four. Determinants(a)
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298 Chapter Four. Determinantsand t
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300 Chapter Four. DeterminantsWe ar
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306 Chapter Four. Determinants? 3.3
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316 Chapter Four. DeterminantsProof
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318 Chapter Four. Determinantš 1.
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320 Chapter Four. DeterminantsIIIOt
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324 Chapter Four. Determinants(a)(
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Chapter FiveSimilarityWhile studyin
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Section I. Complex Vector Spaces 34
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Page 431 and 432: Bibliography[Abbot] Edwin Abbott, F
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Page 435 and 436: [Programmer’s Ref.] Microsoft Pro
Page 437 and 438: composition, A-9self, 361computer a
Page 439 and 440: Kirchhoff’s Laws, 73Klein, F., 28
Page 441 and 442: at infinity, 337in projective plane
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Linear Algebra

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