Linear Algebra

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$Math 223 â Vector Calculus Practice Exam 1 Instructor: Arlo Caine ...$

$Math 212 Review Problems for Test 2$

286 Chapter Three. Maps Between Spacesdecided, it is final, so once in s A or s B , the learner stays there. For the other statechanges, we can posit transitions with probability p in either direction.(a) Construct the transition matrix.(b) Take p = 0.25 and take the initial vector to be 1 at s U . Run this for five steps.What is the chance of ending up at s A ?(c) Do the same for p = 0.20.(d) Graph p versus the chance of ending at s A . Is there a threshold value for p,above which the learner is almost sure not to take longer than five steps?5 A certain town is in a certain country (this is a hypothetical problem). Each yearten percent of the town dwellers move to other parts of the country. Each year onepercent of the people from elsewhere move to the town. Assume that there are twostates s T , living in town, and s C , living elsewhere.(a) Construct the transition matrix.(b) Starting with an initial distribution s T = 0.3 and s C = 0.7, get the results forthe first ten years.(c) Do the same for s T = 0.2.(d) Are the two outcomes alike or different?6 For the World Series application, use a computer to generate the seven vectors forp = 0.55 and p = 0.6.(a) What is the chance of the National League team winning it all, even thoughthey have only a probability of 0.45 or 0.40 of winning any one game?(b) Graph the probability p against the chance that the American League teamwins it all. Is there a threshold value — a p above which the better team isessentially ensured of winning?7 Above we define a transition matrix to have each entry nonnegative and eachcolumn sum to 1.(a) Check that the three transition matrices shown in this Topic meet these twoconditions. Must any transition matrix do so?(b) Observe that if A⃗v 0 = ⃗v 1 and A⃗v 1 = ⃗v 2 then A 2 is a transition matrix from⃗v 0 to ⃗v 2 . Show that a power of a transition matrix is also a transition matrix.(c) Generalize the prior item by proving that the product of two appropriatelysizedtransition matrices is a transition matrix.Computer CodeThis script markov.m for the computer algebra system Octave generated thetable of World Series outcomes. (The hash character # marks the rest of a lineas a comment.)# Octave script file to compute chance of World Series outcomes.function w = markov(p,v)q = 1-p;A=[0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-0p,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-0q,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-1_0,p,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-00,q,p,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-10,0,q,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-2__0,0,0,p,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 3-00,0,0,q,p,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-10,0,0,0,q,p, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-2_0,0,0,0,0,q, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 0-30,0,0,0,0,0, p,0,0,0,1,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 4-00,0,0,0,0,0, q,p,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 3-1__0,0,0,0,0,0, 0,q,p,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 2-20,0,0,0,0,0, 0,0,q,p,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0; # 1-3
Topic: Markov Chains 2870,0,0,0,0,0, 0,0,0,q,0,0, 0,0,1,0,0,0, 0,0,0,0,0,0; # 0-4_0,0,0,0,0,0, 0,0,0,0,0,p, 0,0,0,1,0,0, 0,0,0,0,0,0; # 4-10,0,0,0,0,0, 0,0,0,0,0,q, p,0,0,0,0,0, 0,0,0,0,0,0; # 3-20,0,0,0,0,0, 0,0,0,0,0,0, q,p,0,0,0,0, 0,0,0,0,0,0; # 2-3__0,0,0,0,0,0, 0,0,0,0,0,0, 0,q,0,0,0,0, 1,0,0,0,0,0; # 1-40,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,p,0, 0,1,0,0,0,0; # 4-20,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,q,p, 0,0,0,0,0,0; # 3-3_0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,q, 0,0,0,1,0,0; # 2-40,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,p,0,1,0; # 4-30,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,q,0,0,1]; # 3-4w = A * v;endfunctionThen the Octave session was this.> v0=[1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0]> p=.5> v1=markov(p,v0)> v2=markov(p,v1)...Translating to another computer algebra system should be easy — all havecommands similar to these.
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Jim Hefferonhttp://joshua.smcvt.edu
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PrefaceThis book helps students to
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If you are reading this on your own
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ContentsChapter One: Linear Systems
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Chapter Five: SimilarityI Complex V
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2 Chapter One. Linear Systemsmust e
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4 Chapter One. Linear SystemsEach o
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6 Chapter One. Linear Systems(Had t
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8 Chapter One. Linear Systemsany so
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10 Chapter One. Linear Systems(b) C
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12 Chapter One. Linear SystemsCompa
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14 Chapter One. Linear SystemsMatri
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16 Chapter One. Linear Systems2.12
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18 Chapter One. Linear Systemsthe g
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20 Chapter One. Linear Systems(b) a
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22 Chapter One. Linear SystemsStudy
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24 Chapter One. Linear Systemsleadi
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26 Chapter One. Linear SystemsThat
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28 Chapter One. Linear SystemsWe ha
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30 Chapter One. Linear Systemš 3.
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32 Chapter One. Linear SystemsIILin
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34 Chapter One. Linear Systemsvecto
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36 Chapter One. Linear Systemsand l
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38 Chapter One. Linear Systems(b) t
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40 Chapter One. Linear Systemsis th
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42 Chapter One. Linear Systems2.6 C
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44 Chapter One. Linear Systems(b) S
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46 Chapter One. Linear SystemsIIIRe
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48 Chapter One. Linear Systems1.4 E
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50 Chapter One. Linear SystemsExerc
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52 Chapter One. Linear Systems2.3 L
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54 Chapter One. Linear SystemsBy th
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56 Chapter One. Linear SystemsThe u
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58 Chapter One. Linear Systems2.22
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60 Chapter One. Linear Systems7 2 5
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62 Chapter One. Linear Systemsestim
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64 Chapter One. Linear Systems3 Thi
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66 Chapter One. Linear Systemscompu
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68 Chapter One. Linear Systemsworke
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70 Chapter One. Linear SystemsCompo
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72 Chapter One. Linear SystemsKirch
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74 Chapter One. Linear Systems(b) L
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76 Chapter Two. Vector SpacesIDefin
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78 Chapter Two. Vector SpacesThe ni
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80 Chapter Two. Vector Spaces1.7 De
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82 Chapter Two. Vector Spaces1.13 E
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84 Chapter Two. Vector SpacesLinear
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86 Chapter Two. Vector Spaces1.32 P
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88 Chapter Two. Vector Spacesand sc
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90 Chapter Two. Vector Spacescombin
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94 Chapter Two. Vector Spaceš 2.2
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96 Chapter Two. Vector Spaces(b) Wh
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98 Chapter Two. Vector Spaces1.2 De
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102 Chapter Two. Vector Spaces1.13
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104 Chapter Two. Vector SpacesThus
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106 Chapter Two. Vector Spaceš 1.
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108 Chapter Two. Vector Spaceslinea
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110 Chapter Two. Vector SpacesThe v
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112 Chapter Two. Vector Spacesholds
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116 Chapter Two. Vector Spaces2.2 R
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120 Chapter Two. Vector Spaceš 2.
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122 Chapter Two. Vector Spaces3.4 L
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124 Chapter Two. Vector SpacesThe c
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126 Chapter Two. Vector SpacesProof
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128 Chapter Two. Vector Spaces(c) P
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130 Chapter Two. Vector Spacesthe y
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132 Chapter Two. Vector SpacesFinal
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134 Chapter Two. Vector SpacesIn th
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138 Chapter Two. Vector Spacesusual
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140 Chapter Two. Vector Spacestake
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142 Chapter Two. Vector Spaces(c) F
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144 Chapter Two. Vector SpacesThe i
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146 Chapter Two. Vector Spacesdirec
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148 Chapter Two. Vector Spaces(d) C
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150 Chapter Two. Vector Spaceswill
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152 Chapter Two. Vector SpacesThe s
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154 Chapter Two. Vector SpacesThere
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156 Chapter Two. Vector Spaces
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158 Chapter Three. Maps Between Spa
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Page 246 and 247: 236 Chapter Three. Maps Between Spa
Page 276 and 277: TopicLine of Best FitThis Topic req
Page 292 and 293: TopicMarkov ChainsHere is a simple
Page 298 and 299: TopicOrthonormal MatricesIn The Ele
Page 306 and 307: 296 Chapter Four. DeterminantsIDefi
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Page 320 and 321: 310 Chapter Four. Determinants3.10
Page 322 and 323: 312 Chapter Four. Determinants3.20
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Page 326 and 327: 316 Chapter Four. DeterminantsProof
Page 328 and 329: 318 Chapter Four. DeterminantsFor p
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Page 336 and 337: 326 Chapter Four. DeterminantsIIILa
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336 Chapter Four. DeterminantsCount
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338 Chapter Four. DeterminantsLet C
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340 Chapter Four. Determinants3 The
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342 Chapter Four. DeterminantsIt is
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344 Chapter Four. DeterminantsPSIFo
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346 Chapter Four. Determinantsnonze
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348 Chapter Four. DeterminantsOT 1U
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350 Chapter Four. Determinantsthe c
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352 Chapter Four. Determinants(d) F
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354 Chapter Five. Similaritythe sca
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356 Chapter Five. SimilarityIn C we
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358 Chapter Five. SimilarityIISimil
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360 Chapter Five. Similarity(b) Fin
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362 Chapter Five. Similarity2.4 Lem
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364 Chapter Five. Similarity( ) ( )
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366 Chapter Five. Similaritythen 2
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368 Chapter Five. Similarity3.11 De
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370 Chapter Five. Similarity⃗0 =
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372 Chapter Five. Similarity3.43 Di
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374 Chapter Five. Similarityand thi
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376 Chapter Five. SimilarityThis gr
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378 Chapter Five. SimilarityThis is
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380 Chapter Five. SimilarityProof S
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382 Chapter Five. SimilarityProof F
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384 Chapter Five. Similarity2.17 Ex
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386 Chapter Five. Similarity̌ 2.26
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388 Chapter Five. Similaritywith re
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390 Chapter Five. Similaritycheck t
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392 Chapter Five. SimilarityWe refe
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394 Chapter Five. Similarity1.18 Wh
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396 Chapter Five. Similarityis (x
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398 Chapter Five. Similarity⃗m
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400 Chapter Five. Similarityeach te
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402 Chapter Five. Similarity2.14 Co
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404 Chapter Five. SimilaritySo the
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406 Chapter Five. Similarity(b) The
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TopicMethod of PowersIn application
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410 Chapter Five. SimilarityExercis
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TopicStable PopulationsImagine a re
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TopicPage RankingImagine that you a
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416 Chapter Five. SimilaritySo we r
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TopicLinear RecurrencesIn 1202 Leon
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420 Chapter Five. Similaritya n−k
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422 Chapter Five. Similarityof the
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424 Chapter Five. Similarity(b) f(n
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AppendixMathematics is made of argu
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A-3There are two main ways to estab
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A-5induction. Such a proof has two
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A-7Because of Extensionality, to pr
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A-9same role with respect to functi
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A-11S −1 = {. . . , −3, −1, 1
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[Arrow] Kenneth J. Arrow, Social Ch
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[Giordano, Jaye, Weir] Frank R. Gio
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[Quine] W. V. Quine, Methods of Log
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Indexaccuracyof Gauss’s Method, 6
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Euclid, 288even functions, 94, 134e
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Google, 416identity, 219, 223incide
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of a matrix, 193of a vector, 112rep
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spin, 145well-defined, A-8Wheatston
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Linear Algebra

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