Linear Algebra, 2020a

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[Gardner, 1970] Martin Gardner, Mathematical Games, Some mathematical curiosities embedded in the solar system, Scientific American, April 1970, p. 108 – 112. [Gardner, 1980] Martin Gardner, Mathematical Games, From counting votes to making votes count: the mathematics of elections, Scientific American, October 1980. [Gardner, 1974] Martin Gardner, Mathematical Games, On the paradoxical situations that arise from nontransitive relations, Scientific American, October 1974. [Giordano, Wells, Wilde] Frank R. Giordano, Michael E. Wells, Carroll O. Wilde, Dimensional Analysis, UMAP Unit 526, in UMAP Modules, 1987, COMAP, 1987. [Giordano, Jaye, Weir] Frank R. Giordano, Michael J. Jaye, Maurice D. Weir, The Use of Dimensional Analysis in Mathematical Modeling, UMAP Unit 632, in UMAP Modules, 1986, COMAP, 1986. [Google Maps] Directions — Google Maps, http://maps.google.com/help/maps/directions/, 2012-Jan-07. [Goult, et al.] R.J. Goult, R.F. Hoskins, J.A. Milner, M.J. Pratt, Computational Methods in Linear Algebra, Wiley, 1975. [Graham, Knuth, Patashnik] Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1988. [Halmos] Paul R. Halmos, Finite Dimensional Vector Spaces, second edition, Van Nostrand, 1958. [Hamming] Richard W. Hamming, Introduction to Applied Numerical Analysis, Hemisphere Publishing, 1971. [Hanes] Kit Hanes, Analytic Projective Geometry and its Applications, UMAP Unit 710, UMAP Modules, 1990, p. 111. [Heath] T. Heath, Euclid’s Elements, volume 1, Dover, 1956. [Hefferon] J Hefferon, Introduction to Proofs, an Inquiry-Based approach, http://joshua.smcvt.edu/proofs/, 2013. [Hoffman & Kunze] Kenneth Hoffman, Ray Kunze, Linear Algebra, second edition, Prentice-Hall, 1971. [Hofstadter] Douglas R. Hofstadter, Metamagical Themas: Questing for the Essence of Mind and Pattern, Basic Books, 1985. [Hughes et al.] John F. Hughes, Andries van Dam, Morgan McGuire, David F. Sklar, James D. Foley, Steven K. Feiner, Kurt Akeley, Computer graphics: principles and practice, third edition, Addison-Wesley, 1995. [Iosifescu] Marius Iofescu, Finite Markov Processes and Their Applications, John Wiley, 1980.
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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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Section IV. Jordan Form 449 Convers
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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 510 and 511: [Am. Math. Mon., Dec. 1966] Hans Li
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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