Linear Algebra, 2020a

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[Am. Math. Mon., Dec. 1966] Hans Liebeck, A Proof of the Equality of Column Rank and Row Rank of a Matrix American Mathematical Monthly, vol. 73 no. 10 (Dec. 1966), p. 1114. [Anton] Howard Anton, Elementary Linear Algebra, John Wiley & Sons, 1987. [Arrow] Kenneth J. Arrow, Social Choice and Individual Values, Wiley, 1963. [Austin] David Austin, How Google Finds Your Needle in the Web’s Haystack, http://www.ams.org/samplings/feature-column/fcarc-pagerank, retrieved Feb. 2012. [Ball & Coxeter] W.W. Rouse Ball, Mathematical Recreations and Essays, revised by H.S.M. Coxeter, MacMillan, 1962. [Beck] Matthias Beck, Ross Geoghegan, The Art of Proof, http://math.sfsu.edu/beck/papers/aop.noprint.pdf, 2011-Aug-08. [Beardon] A.F. Beardon, The Dimension of the Space of Magic Squares, The Mathematical Gazette, vol. 87, no. 508 (Mar. 2003), p. 112-114. [Berg & Marshall] Richar E Berg, Todd S Marshall, Wilberforce pendulum oscillations and normal modes, American Journal of Physics, volume 59 number 1 (Jan 1991), p. 32–38. [Birkhoff & MacLane] Garrett Birkhoff, Saunders MacLane, Survey of Modern Algebra, third edition, Macmillan, 1965. [Blass 1984] A. Blass, Existence of Bases Implies the Axiom of Choice, pp. 31 – 33, Axiomatic Set Theory, J. E. Baumgartner, ed., American Mathematical Society, Providence RI, 1984. [Bridgman] P.W. Bridgman, Dimensional Analysis, Yale University Press, 1931. [Brin & Page] Sergey Brin and Lawrence Page, The Anatomy of a Large-Scale Hypertextual Web Search Engine, http://infolab.stanford.edu/pub/papers/google.pdf, retrieved Feb. 2012. [Bryan & Leise] Kurt Bryan, Tanya Leise, The $25,000,000,000 Eigenvector: the Linear Algebra Behind Google, SIAM Review, Vol. 48, no. 3 (2006), p. 569-81. [Casey] John Casey, The Elements of Euclid, Books I to VI and XI, ninth edition, Hodges, Figgis, and Co., Dublin, 1890. [ChessMaster] User ChessMaster of StackOverflow, answer to Python determinant calculation(without the use of external libraries), http://stackoverflow.com/a/10037087/238366, answer posted 2012-Apr-05, accessed 2012-Jun-18. [Chicken Scheme] Free software implementation, Felix L. Winkelmann and The Chicken Team, http://wiki.call-cc.org/, accessed 2013-Nov-20. [Clark & Coupe] David H. Clark, John D. Coupe, The Bangor Area Economy Its Present and Future, report to the city of Bangor ME, Mar. 1967. [Cleary] R. Cleary, private communication, Nov. 2011.
[Clarke] Arthur C. Clarke, Technical Error, Fantasy, December 1946, reprinted in Great SF Stories 8 (1946), DAW Books, 1982. [Con. Prob. 1955] The Contest Problem Book, 1955 number 38. [Cost Of Tolls] Cost of Tolls, http://costoftolls.com/Tolls_in_New_York.html, 2012-Jan-07. [Coxeter] H.S.M. Coxeter, Projective Geometry, second edition, Springer-Verlag, 1974. [Courant & Robbins] Richard Courant, Herbert Robbins, What is Mathematics?, Oxford University Press, 1978. [Cullen] Charles G. Cullen, Matrices and Linear Transformations, second edition, Dover, 1990. [Dalal, et. al.] Siddhartha R. Dalal, Edward B. Fowlkes, & Bruce Hoadley, Lesson Learned from Challenger: A Statistical Perspective, Stats: the Magazine for Students of Statistics, Fall 1989, p. 3. [Davies] Thomas D. Davies, New Evidence Places Peary at the Pole, National Geographic Magazine, vol. 177 no. 1 (Jan. 1990), p. 44. [de Mestre] Neville de Mestre, The Mathematics of Projectiles in Sport, Cambridge University Press, 1990. [De Parville] De Parville, La Nature, Paris, 1884, part I, p. 285 – 286 (citation from [Ball & Coxeter]). [Ebbing] Darrell D. Ebbing, General Chemistry, fourth edition, Houghton Mifflin, 1993. [Ebbinghaus] H. D. Ebbinghaus, Numbers, Springer-Verlag, 1990. [Einstein] A. Einstein, Annals of Physics, v. 35, 1911, p. 686. [Eggar] M.H. Eggar, Pinhole Cameras, Perspective, and Projective Geometry, American Mathematical Monthly, August-September 1998, p. 618 – 630. [Falbo] Clement Falbo, The Golden Ratio — a Contrary Viewpoint, College Mathematics Journal, vol. 36, no. 2, March 2005, p. 123 – 134. [Feller] William Feller, An Introduction to Probability Theory and Its Applications (vol. 1, 3rd ed.), John Wiley, 1968. [Fuller & Logan] L.E. Fuller & J.D. Logan, On the Evaluation of Determinants by Chiò’s Method, p 49-52, in Linear Algebra Gems, Carlson, et al, Mathematical Association of America, 2002. [Gardner] Martin Gardner, The New Ambidextrous Universe, third revised edition, W. H. Freeman and Company, 1990. [Gardner 1957] Martin Gardner, Mathematical Games: About the remarkable similarity between the Icosian Game and the Tower of Hanoi, Scientific American, May 1957, p. 150 – 154.
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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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20 Chapter One. Linear Systems We
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22 Chapter One. Linear Systems reve
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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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Section III. Basis and Dimension 12
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Topic Fields Computations involving
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Topic Crystals Everyone has noticed
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Topic: Crystals 157 is a good basis
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Topic Voting Paradoxes Imagine that
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Topic: Voting Paradoxes 161 subspac
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Topic: Voting Paradoxes 163 between
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Topic Dimensional Analysis “You c
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Topic: Dimensional Analysis 167 for
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Topic: Dimensional Analysis 169 whi
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Topic: Dimensional Analysis 171 (b)
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Chapter Three Maps Between Spaces I
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Section I. Isomorphisms 175 and sca
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Section I. Isomorphisms 177 The fir
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Section I. Isomorphisms 179 picture
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Section I. Isomorphisms 181 (b) f:
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Section I. Isomorphisms 183 (d) Pro
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Section I. Isomorphisms 185 2.3 The
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Section I. Isomorphisms 187 In the
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Section I. Isomorphisms 189 Associa
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Section II. Homomorphisms 191 II Ho
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Section II. Homomorphisms 193 So on
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Section II. Homomorphisms 195 1.12
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Section II. Homomorphisms 201 Recal
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Section II. Homomorphisms 203 Now,
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Section II. Homomorphisms 207 Exerc
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Section II. Homomorphisms 209 (a) h
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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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Page 475 and 476: Topic: Method of Powers 465 ⃗v T
Page 477 and 478: Topic: Method of Powers 467 39368 -
Page 479 and 480: Topic: Stable Populations 469 Now i
Page 481 and 482: Topic: Page Ranking 471 importance
Page 483 and 484: Topic: Page Ranking 473 Exercises 1
Page 485 and 486: Topic: Linear Recurrences 475 Write
Page 487 and 488: Topic: Linear Recurrences 477 We ca
Page 489 and 490: Topic: Linear Recurrences 479 place
Page 491 and 492: Topic: Linear Recurrences 481 After
Page 493 and 494: Topic: Coupled Oscillators 483 We s
Page 495 and 496: Topic: Coupled Oscillators 485 we a
Page 497 and 498: Appendix Mathematics is made of arg
Page 499 and 500: A-3 hypothesis. This argument is wr
Page 501 and 502: A-5 Another example is this proof t
Page 503 and 504: A-7 A collection that is like a set
Page 505 and 506: A-9 A function has a left inverse i
Page 507: A-11 related to 102. Verifying the
Page 512 and 513: [Gardner, 1970] Martin Gardner, Mat
Page 514 and 515: [Online Encyclopedia of Integer Seq
Page 516 and 517: Index accuracy of Gauss’s Method,
Page 518 and 519: elementary reduction operations, 5
Page 520 and 521: linear independence multiset, 113 l
Page 522 and 523: in projective plane, 383, 392 polyn
Page 524 and 525: summation notation, for permutation
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