Linear Algebra

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

$Math 212 Review Problems for Test 2$

science in general, and of Linear Algebra in particular. (I took the liberty offormatting them as poetry).I know of no better tacticthan the illustration of exciting principlesby well-chosen particulars.–Stephen Jay GouldIf you really wish to learnthen you must mount the machineand become acquainted with its tricksby actual trial.–Wilbur WrightJim HefferonMathematics, Saint Michael’s CollegeColchester, Vermont USA 05439http://joshua.smcvt.edu/linearalgebra2012-Feb-29Author’s Note. Inventing a good exercise, one that enlightens as well as tests,is a creative act and hard work. The inventor deserves recognition. But textshave traditionally not given attributions for questions. I have changed that herewhere I was sure of the source. I would be glad to hear from anyone who canhelp me to correctly attribute others of the questions.– iv –
ContentsChapter One: Linear SystemsI Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 1I.1 Gauss’s Method . . . . . . . . . . . . . . . . . . . . . . . . . 2I.2 Describing the Solution Set . . . . . . . . . . . . . . . . . . . 11I.3 General = Particular + Homogeneous . . . . . . . . . . . . . . 20II Linear Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 32II.1 Vectors in Space* . . . . . . . . . . . . . . . . . . . . . . . . 32II.2 Length and Angle Measures* . . . . . . . . . . . . . . . . . . 39III Reduced Echelon Form . . . . . . . . . . . . . . . . . . . . . . . . 46III.1 Gauss-Jordan Reduction . . . . . . . . . . . . . . . . . . . . . 46III.2 The Linear Combination Lemma . . . . . . . . . . . . . . . . 51Topic: Computer Algebra Systems . . . . . . . . . . . . . . . . . . . 59Topic: Input-Output Analysis . . . . . . . . . . . . . . . . . . . . . . 61Topic: Accuracy of Computations . . . . . . . . . . . . . . . . . . . . 65Topic: Analyzing Networks . . . . . . . . . . . . . . . . . . . . . . . . 69Chapter Two: Vector SpacesI Definition of Vector Space . . . . . . . . . . . . . . . . . . . . . . 76I.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 76I.2 Subspaces and Spanning Sets . . . . . . . . . . . . . . . . . . 87II Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . 97II.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . 97III Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . . . . 109III.1 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109III.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115III.3 Vector Spaces and Linear Systems . . . . . . . . . . . . . . . 121III.4 Combining Subspaces* . . . . . . . . . . . . . . . . . . . . . . 128Topic: Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Topic: Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139Topic: Voting Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . 143Topic: Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . 149
Page 1 and 2: Jim Hefferonhttp://joshua.smcvt.edu
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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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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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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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316 Chapter Four. DeterminantsProof
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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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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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372 Chapter Five. Similarity3.43 Di
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374 Chapter Five. Similarityand thi
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380 Chapter Five. SimilarityProof S
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382 Chapter Five. SimilarityProof F
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402 Chapter Five. Similarity2.14 Co
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TopicMethod of PowersIn application
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TopicStable PopulationsImagine a re
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TopicPage RankingImagine that you a
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TopicLinear RecurrencesIn 1202 Leon
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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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