Linear Algebra

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

$Math 212 Review Problems for Test 2$

416 Chapter Five. SimilaritySo we rank the first and fourth pages as of equal importance. We rank thesecond and third pages as much more important than those, and about equal inimportance as each other.We’ll add one more refinement. We will allow the surfer to pick a newpage at random even if they are not on a dangling page. Let this happen withprobability α.⎛⎞⎛⎞0 0 1/3 1/41/4 1/4 1/4 1/41 0 1/3 1/4G = α · ⎜⎟⎝0 1 0 1/4⎠ + (1 − α) · 1/4 1/4 1/4 1/4⎜⎟⎝1/4 1/4 1/4 1/4⎠0 0 1/3 1/41/4 1/4 1/4 1/4This is the Google matrix .In practice α is typically between 0.85 and 0.99. Here are the ranks for thefour pages with a spread of α’s.α 0.85 0.90 0.95 0.99p 1 0.325 0.317 0.309 0.302p 2 0.602 0.602 0.602 0.601p 3 0.652 0.661 0.669 0.675p 4 0.325 0.317 0.309 0.302The details of the algorithms used by commercial search engines are secret,no doubt have many refinements, and also change frequently. But theinventors of Google were gracious enough to outline the basis for their work in[Brin & Page]. A more current source is [Wikipedia Google Page Rank]. Twoadditional excellent expositions are [Wills] and [Austin].Exercises1 A square matrix is stochastic if the sum of the entries in each column is one. TheGoogle matrix is computed by taking a combination G = α ∗ H + (1 − α) ∗ S of twostochastic matrices. Show that G must be stochastic.2 For this web of pages, the importance of each page should be equal. Verify it forα = 0.85.p 1 p 2p 4p 33 [Bryan & Leise] Give the importance ranking for this web of pages.p 1 p 2p 4p 3
Topic: Page Ranking 417(a) Use α = 0.85.(b) Use α = 0.95.(c) Observe that while p 3 is linked-to from all other pages, and therefore seemsimportant, it is not the highest ranked page. What is the highest ranked page?Explain.
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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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TopicLine of Best FitThis Topic req
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TopicMarkov ChainsHere is a simple
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TopicOrthonormal MatricesIn The Ele
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296 Chapter Four. DeterminantsIDefi
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298 Chapter Four. Determinantsalso
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300 Chapter Four. Determinantš 1.
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302 Chapter Four. DeterminantsThe s
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304 Chapter Four. Determinants∣
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306 Chapter Four. Determinantsdeter
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308 Chapter Four. DeterminantsSo wi
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310 Chapter Four. Determinants3.10
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312 Chapter Four. Determinants3.20
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314 Chapter Four. Determinantsis: t
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316 Chapter Four. DeterminantsProof
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318 Chapter Four. DeterminantsFor p
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320 Chapter Four. DeterminantsIIGeo
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324 Chapter Four. Determinantš 1.
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326 Chapter Four. DeterminantsIIILa
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328 Chapter Four. DeterminantsAlter
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330 Chapter Four. Determinants⎛
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TopicSpeed of Calculating Determina
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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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Page 390 and 391: 380 Chapter Five. SimilarityProof S
Page 392 and 393: 382 Chapter Five. SimilarityProof F
Page 394 and 395: 384 Chapter Five. Similarity2.17 Ex
Page 396 and 397: 386 Chapter Five. Similarity̌ 2.26
Page 398 and 399: 388 Chapter Five. Similaritywith re
Page 400 and 401: 390 Chapter Five. Similaritycheck t
Page 402 and 403: 392 Chapter Five. SimilarityWe refe
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Page 416 and 417: 406 Chapter Five. Similarity(b) The
Page 418 and 419: TopicMethod of PowersIn application
Page 420 and 421: 410 Chapter Five. SimilarityExercis
Page 422 and 423: TopicStable PopulationsImagine a re
Page 424 and 425: TopicPage RankingImagine that you a
Page 428 and 429: TopicLinear RecurrencesIn 1202 Leon
Page 430 and 431: 420 Chapter Five. Similaritya n−k
Page 432 and 433: 422 Chapter Five. Similarityof the
Page 434 and 435: 424 Chapter Five. Similarity(b) f(n
Page 436 and 437: AppendixMathematics is made of argu
Page 438 and 439: A-3There are two main ways to estab
Page 440 and 441: A-5induction. Such a proof has two
Page 442 and 443: A-7Because of Extensionality, to pr
Page 444 and 445: A-9same role with respect to functi
Page 446: A-11S −1 = {. . . , −3, −1, 1
Page 449 and 450: [Arrow] Kenneth J. Arrow, Social Ch
Page 451 and 452: [Giordano, Jaye, Weir] Frank R. Gio
Page 453 and 454: [Quine] W. V. Quine, Methods of Log
Page 455 and 456: Indexaccuracyof Gauss’s Method, 6
Page 457 and 458: Euclid, 288even functions, 94, 134e
Page 459 and 460: Google, 416identity, 219, 223incide
Page 461 and 462: of a matrix, 193of a vector, 112rep
Page 463: spin, 145well-defined, A-8Wheatston
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Linear Algebra

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