Linear Algebra, 2020a

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162 Chapter Two. Vector Spaces Thus we can see that this D>R>T voter’s rational preference list does have a cyclic part. The T>R>Dvoter is opposite to the one just considered in that the ‘>’ symbols are reversed. This voter’s decomposition 1 T D R −1 = −1/3 T D R −1/3 + 4/3 T D R −2/3 −1 −1/3 −2/3 shows that these opposite preferences have decompositions that are opposite. We say that the first voter has positive spin since the cycle part is with the direction that we have chosen for the arrows, while the second voter’s spin is negative. The fact that these opposite voters cancel each other is reflected in the fact that their vote vectors add to zero. This suggests an alternate way to tally an election. We could first cancel as many opposite preference lists as possible and then determine the outcome by adding the remaining lists. The table below contains the three pairs of opposite preference lists. For instance, the top line contains the voters discussed above. positive spin negative spin −1 T Democrat > Republican > Third D R 1 = D D 1/3 1/3 −4/3 T R + T R 2/3 1 T Third > Republican > Democrat D R −1 = D D −1/3 −1/3 4/3 T R + T R −2/3 1 1/3 2/3 −1 −1/3 −2/3 1 T Republican > Third > Democrat D R −1 = D D 1/3 1/3 2/3 T R + T R −4/3 −1 T Democrat > Third > Republican D R 1 = D D −1/3 −1/3 −2/3 T R + T R 4/3 1 1/3 2/3 −1 −1/3 −2/3 1 T Third > Democrat > Republican D R 1 = D D 1/3 1/3 2/3 T R + T R 2/3 −1 T Republican > Democrat > Third D R −1 = D D −1/3 −1/3 −2/3 T R + T R −2/3 −1 1/3 −4/3 1 −1/3 4/3 If we conduct the election as just described then after the cancellation of as many opposite pairs of voters as possible there will remain three sets of preference lists: one set from the first row, one from the second row, and one from the third row. We will finish by proving that a voting paradox can happen only if the spins of these three sets are in the same direction. That is, for a voting paradox to occur the three remaining sets must all come from the left of the table or all come from the right (see Exercise 3). This shows that there is some connection
Topic: Voting Paradoxes 163 between the majority cycle and the decomposition that we are using — a voting paradox can happen only when the tendencies toward cyclic preference reinforce each other. For the proof, assume that we have canceled opposite preference orders and are left with one set of preference lists for each of the three rows. Consider the first row’s remaining preference lists. They could be from the first row’s left or right (or between, since the lists could have canceled exactly). We shall write −a D a T a where a is an integer that is positive if the remaining lists are on the left, where a is negative if the lists are on the right, and zero if the cancellation was perfect. Similarly we have integers b and c for the second and third rows, which can each be positive, negative, or zero. Then the election is determined by this sum. −a T D a R a + b T D b R −b + c T R D −c R c = −a + b + c T D R a + b − c a − b + c A voting paradox occurs when the three numbers in the total cycle on the right, −a + b + c and a − b + c and a + b − c, are all nonnegative or all nonpositive. We will prove this occurs only when either all three of a, b, and c are nonnegative or all three are nonpositive. Let the total cycle numbers be nonnegative; the other case is similar. −a + b + c 0 a − b + c 0 a + b − c 0 Add the first two rows to see that c 0. Add the first and third rows for b 0. And, the second and third rows together give a 0. Thus if the total cycle is nonnegative then in each row the remaining preference lists are from the table’s left. That ends the proof. This result says only that having all three spin in the same direction is a necessary condition for a majority cycle. It is not sufficient; see Exercise 4. Voting theory and associated topics are the subject of current research. There are many intriguing results, notably those produced by K Arrow [Arrow] who won the Nobel Prize in part for this work, showing that no voting system is entirely fair (for a reasonable definition of “fair”). Some good introductory articles are [Gardner, 1970], [Gardner, 1974], [Gardner, 1980], and [Neimi & Riker]. [Taylor] is a readable text. The long list of cases from recent American political
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LINEAR ALGEBRA Jim Hefferon Fourth
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Preface This book helps students to
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Advice. This book’s emphasis on m
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Contents Chapter One: Linear System
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III Laplace’s Formula ...........
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2 Chapter One. Linear Systems must
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4 Chapter One. Linear Systems 1.5 T
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6 Chapter One. Linear Systems 1.9 E
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8 Chapter One. Linear Systems 1.14
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10 Chapter One. Linear Systems ̌ 1
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14 Chapter One. Linear Systems free
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16 Chapter One. Linear Systems abov
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18 Chapter One. Linear Systems We w
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26 Chapter One. Linear Systems eche
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28 Chapter One. Linear Systems To f
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30 Chapter One. Linear Systems Proo
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32 Chapter One. Linear Systems than
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34 Chapter One. Linear Systems (a)
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36 Chapter One. Linear Systems to f
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38 Chapter One. Linear Systems The
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40 Chapter One. Linear Systems line
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42 Chapter One. Linear Systems ̌ 1
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44 Chapter One. Linear Systems Note
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50 Chapter One. Linear Systems III
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52 Chapter One. Linear Systems elem
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54 Chapter One. Linear Systems One
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56 Chapter One. Linear Systems III.
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58 Chapter One. Linear Systems We n
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60 Chapter One. Linear Systems and
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66 Chapter One. Linear Systems [0,0
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68 Chapter One. Linear Systems Now
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70 Chapter One. Linear Systems (b)
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Topic Accuracy of Computations Gaus
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74 Chapter One. Linear Systems The
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Topic Analyzing Networks The diagra
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78 Chapter One. Linear Systems and
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80 Chapter One. Linear Systems (c)
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Chapter Two Vector Spaces The first
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Section I. Definition of Vector Spa
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Section II. Linear Independence 109
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Page 177 and 178: Topic: Dimensional Analysis 167 for
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Page 183 and 184: Chapter Three Maps Between Spaces I
Page 185 and 186: Section I. Isomorphisms 175 and sca
Page 187 and 188: Section I. Isomorphisms 177 The fir
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Page 203 and 204: Section II. Homomorphisms 193 So on
Page 205 and 206: Section II. Homomorphisms 195 1.12
Page 211 and 212: Section II. Homomorphisms 201 Recal
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Page 217 and 218: Section II. Homomorphisms 207 Exerc
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Section III. Computing Linear Maps
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Section IV. Matrix Operations 233 1
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Section IV. Matrix Operations 235 (
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Section IV. Matrix Operations 237 D
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Section IV. Matrix Operations 239 A
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Section IV. Matrix Operations 241 e
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Section IV. Matrix Operations 245 T
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Section IV. Matrix Operations 247 a
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Section IV. Matrix Operations 251 a
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Section IV. Matrix Operations 253 R
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Section IV. Matrix Operations 255 W
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Section V. Change of Basis 263 1.2
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Section V. Change of Basis 265 to t
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Section V. Change of Basis 267 ̌ 1
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Section V. Change of Basis 269 for
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Section V. Change of Basis 271 beca
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Section V. Change of Basis 273 Exer
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Section VI. Projection 275 VI Proje
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Section VI. Projection 277 1.4 Exam
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Section VI. Projection 279 (a) ( 1
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Section VI. Projection 281 For the
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Section VI. Projection 283 By the f
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Section VI. Projection 285 (c) Let
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Section VI. Projection 287 We will
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Section VI. Projection 289 of the v
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Section VI. Projection 291 is perpe
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Section VI. Projection 293 3.19 Wha
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Topic Line of Best Fit This Topic r
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Topic: Line of Best Fit 297 the sam
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Topic: Line of Best Fit 299 4 Find
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Topic Geometry of Linear Maps These
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Topic: Geometry of Linear Maps 303
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Topic: Magic Squares 309 One interp
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Topic: Magic Squares 311 1 2 3 4 5
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Topic Markov Chains Here is a simpl
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Topic: Markov Chains 315 a middle c
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Topic: Markov Chains 317 3 [Kelton]
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Topic Orthonormal Matrices In The E
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Topic: Orthonormal Matrices 321 (we
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Topic: Orthonormal Matrices 323 ( a
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Chapter Four Determinants In the fi
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Section I. Definition 327 A good st
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Section I. Definition 329 the opera
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Section I. Definition 331 is the ve
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Section I. Definition 333 A singula
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Section I. Definition 335 could pos
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Section I. Definition 337 I.3 The P
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Section I. Definition 339 Now this
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Section I. Definition 341 matrix, k
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Section I. Definition 343 Computing
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Section I. Definition 345 3.34 A ma
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Section I. Definition 347 could be
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Section I. Definition 349 Thus, in
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Section I. Definition 351 That is,
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Section I. Definition 353 in the to
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Section II. Geometry of Determinant
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Section III. Laplace’s Formula 36
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Topic Cramer’s Rule A linear syst
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Topic: Cramer’s Rule 371 x − y
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Topic: Speed of Calculating Determi
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Topic: Speed of Calculating Determi
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Topic: Chiò’s Method 377 This re
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Topic: Chiò’s Method 379 (a) 2 1
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Topic: Projective Geometry 381 This
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Topic: Projective Geometry 383 Ther
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Topic: Projective Geometry 385 (We
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Topic: Projective Geometry 387 the
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Topic: Projective Geometry 389 The
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Topic: Projective Geometry 391 Exer
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Topic: Computer Graphics 393 In thi
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Topic: Computer Graphics 395 In thi
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Chapter Five Similarity We have sho
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Section I. Complex Vector Spaces 39
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Section I. Complex Vector Spaces 40
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Section II. Similarity 403 represen
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Section II. Similarity 405 A common
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Section II. Similarity 407 ̌ 1.13
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Section II. Similarity 409 2.4 Lemm
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Section II. Similarity 411 Exercise
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Section II. Similarity 413 where x
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Section II. Similarity 415 We next
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Section II. Similarity 417 3.12 Def
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Section II. Similarity 419 3.18 The
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Section II. Similarity 421 a criter
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Section II. Similarity 423 3.46 Dia
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Section III. Nilpotence 425 has thi
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Section III. Nilpotence 427 1.7 Exa
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Section III. Nilpotence 429 2.2 Cor
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Section III. Nilpotence 431 2.9 Exa
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Section III. Nilpotence 433 But, th
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Section III. Nilpotence 435 (In the
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Section III. Nilpotence 437 2.19 Ex
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Section III. Nilpotence 439 ̌ 2.24
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Section IV. Jordan Form 441 The pol
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Section IV. Jordan Form 443 Proof W
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Section IV. Jordan Form 445 We refe
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Section IV. Jordan Form 447 (a) (e)
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Section IV. Jordan Form 449 Convers
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Section IV. Jordan Form 451 and to
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Section IV. Jordan Form 453 The dia
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Section IV. Jordan Form 455 The rig
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Section IV. Jordan Form 457 2.12 Ex
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Section IV. Jordan Form 459 So the
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Section IV. Jordan Form 461 Therefo
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Section IV. Jordan Form 463 2.38 In
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Topic: Method of Powers 465 ⃗v T
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Topic: Method of Powers 467 39368 -
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Topic: Stable Populations 469 Now i
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Topic: Page Ranking 471 importance
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Topic: Page Ranking 473 Exercises 1
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Topic: Linear Recurrences 475 Write
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Topic: Linear Recurrences 477 We ca
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Topic: Linear Recurrences 479 place
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Topic: Linear Recurrences 481 After
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Topic: Coupled Oscillators 483 We s
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Topic: Coupled Oscillators 485 we a
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Appendix Mathematics is made of arg
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A-3 hypothesis. This argument is wr
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A-5 Another example is this proof t
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A-7 A collection that is like a set
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A-9 A function has a left inverse i
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A-11 related to 102. Verifying the
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[Am. Math. Mon., Dec. 1966] Hans Li
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[Gardner, 1970] Martin Gardner, Mat
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[Online Encyclopedia of Integer Seq
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Index accuracy of Gauss’s Method,
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elementary reduction operations, 5
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linear independence multiset, 113 l
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in projective plane, 383, 392 polyn
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summation notation, for permutation
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