Linear Algebra, 2020a

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354 Chapter Four. Determinants ̌ 4.18 Show that a permutation matrix with m inversions can be row swapped to the identity in m steps. Contrast this with Corollary 4.5. ̌ 4.19 For any permutation φ let g(φ) be the integer defined in this way. g(φ) = ∏ i
Section II. Geometry of Determinants 355 II Geometry of Determinants The prior section develops the determinant algebraically, by considering formulas satisfying certain conditions. This section complements that with a geometric approach. Beyond its intuitive appeal, an advantage of this approach is that while we have so far only considered whether or not a determinant is zero, here we shall give a meaning to the value of the determinant. (The prior section treats the determinant as a function of the rows but this section focuses on columns.) II.1 Determinants as Size Functions This parallelogram picture is familiar from the construction of the sum of the two vectors. ( x2 y 2 ) ( x1 y 1 ) 1.1 Definition In R n the box (or parallelepiped) formed by 〈⃗v 1 ,...,⃗v n 〉 is the set {t 1 ⃗v 1 + ···+ t n ⃗v n | t 1 ,...,t n ∈ [0 ... 1]}. Thus the parallelogram above is the box formed by 〈 ( x 1 y 1 ) , ( x2 y 2 ) 〉. A three-space box is shown in Example 1.4. We can find the area of the above box by drawing an enclosing rectangle and subtracting away areas not in the box. B A y 2 D y 1 C F E x 2 x 1 area of parallelogram = area of rectangle − area of A − area of B − ···− area of F =(x 1 + x 2 )(y 1 + y 2 )−x 2 y 1 − x 1 y 1 /2 − x 2 y 2 /2 − x 2 y 2 /2 − x 1 y 1 /2 − x 2 y 1 = x 1 y 2 − x 2 y 1 That the area equals the value of the determinant ∣ x 1 x ∣∣∣∣ 2 = x ∣ 1 y 2 − x 2 y 1 y 1 y 2
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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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Page 319 and 320: Topic: Magic Squares 309 One interp
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Page 323 and 324: Topic Markov Chains Here is a simpl
Page 325 and 326: Topic: Markov Chains 315 a middle c
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Page 329 and 330: Topic Orthonormal Matrices In The E
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Page 335 and 336: Chapter Four Determinants In the fi
Page 337 and 338: Section I. Definition 327 A good st
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Page 347 and 348: Section I. Definition 337 I.3 The P
Page 349 and 350: Section I. Definition 339 Now this
Page 351 and 352: Section I. Definition 341 matrix, k
Page 353 and 354: Section I. Definition 343 Computing
Page 355 and 356: Section I. Definition 345 3.34 A ma
Page 357 and 358: Section I. Definition 347 could be
Page 359 and 360: Section I. Definition 349 Thus, in
Page 361 and 362: Section I. Definition 351 That is,
Page 363: Section I. Definition 353 in the to
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Page 379 and 380: Topic Cramer’s Rule A linear syst
Page 381 and 382: Topic: Cramer’s Rule 371 x − y
Page 383 and 384: Topic: Speed of Calculating Determi
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Page 387 and 388: Topic: Chiò’s Method 377 This re
Page 389 and 390: Topic: Chiò’s Method 379 (a) 2 1
Page 391 and 392: Topic: Projective Geometry 381 This
Page 393 and 394: Topic: Projective Geometry 383 Ther
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Page 407 and 408: Chapter Five Similarity We have sho
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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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Linear Algebra, 2020a

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