Linear Algebra

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

$Math 212 Review Problems for Test 2$

356 Chapter Five. SimilarityIn C we can factor (obviously, at least some) quadratics that would beirreducible if we were to stick to the real numbers. Surprisingly, in C we cannot only factor x 2 + 1 and its close relatives, we can factor any quadratic.ax 2 + bx + c = a · (x− −b + √ b 2 − 4ac2a) ( −b − √ b 2 − 4ac)· x −2a1.10 Example The second degree polynomial x 2 + x + 1 factors over the complexnumbers into the product of two first degree polynomials.(x −−1 + √ −32)( −1 − √ √−3) ( 1 3x − = x − (−22 + 2 i))( x − (− 1 √32 − 2 i))1.11 Theorem (Fundamental Theorem of <strong>Algebra</strong>) Polynomials with complex coefficientsfactor into linear polynomials with complex coefficients. The factorizationis unique.I.2 Complex RepresentationsRecall the definitions of the complex number additionand multiplication.(a + bi) + (c + di) = (a + c) + (b + d)i(a + bi)(c + di) = ac + adi + bci + bd(−1)= (ac − bd) + (ad + bc)i2.1 Example For instance, (1 − 2i) + (5 + 4i) = 6 + 2i and (2 − 3i)(4 − 0.5i) =6.5 − 13i.Handling scalar operations with those rules, all of the operations that we’vecovered for real vector spaces carry over unchanged.2.2 Example Matrix multiplication is the same, although the scalar arithmeticinvolves more bookkeeping.() ()1 + 1i 2 − 0i 1 + 0i 1 − 0ii(−2 + 3i 3i −i)(1 + 1i) · (1 + 0i) + (2 − 0i) · (3i) (1 + 1i) · (1 − 0i) + (2 − 0i) · (−i)=(i) · (1 + 0i) + 3i) · (3i) (i) · (1 − 0i) + (−2 + 3i) · (−i)(+ (−2)=1 + 7i 1 − 1i−9 − 5i 3 + 3i
Section I. Complex Vector Spaces 357We shall also carry over unchanged from the previous chapters everythingthat we can. For instance, we shall call this⎛ ⎞ ⎛ ⎞1 + 0i 0 + 0i0 + 0i〈⎜ .⎟⎝ . ⎠ , . . . , 0 + 0i⎜ ⎟⎝ . ⎠ 〉0 + 0i 1 + 0ithe standard basis for C n as a vector space over C and again denote it E n .
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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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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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