Linear Algebra, 2020a

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Topic Accuracy of Computations Gauss’s Method lends itself to computerization. The code below illustrates. It operates on an n×n matrix named a, doing row combinations using the first row, then the second row, etc. for(row=1; row
Topic: Accuracy of Computations 73 it will represent the second equation internally as 1.000 000 0x+2y = 3.000 000 0, losing the digits in the ninth place. Instead of reporting the correct solution, this computer will think that the two equations are equal and it will report that the system is singular. For some intuition about how the computer could come up with something that far off, consider this graph of the system. (1, 1) We cannot tell the two lines apart; this system is nearly singular in the sense that the two lines are nearly the same line. This gives the system (∗) the property that a small change in an equation can cause a large change in the solution. For instance, changing the 3.000 000 01 to 3.000 000 03 changes the intersection point from (1, 1) to (3, 0). The solution changes radically depending on the ninth digit, which explains why an eight-place computer has trouble. A problem that is very sensitive to inaccuracy or uncertainties in the input values is ill-conditioned. The above example gives one way in which a system can be difficult to solve on a computer. It has the advantage that the picture of nearly-equal lines gives a memorable insight into one way for numerical difficulties to happen. Unfortunately this insight isn’t useful when we wish to solve some large system. We typically will not understand the geometry of an arbitrary large system. There are other ways that a computer’s results may be unreliable, besides that the angle between some of the linear surfaces is small. For example, consider this system (from [Hamming]). 0.001x + y = 1 x − y = 0 (∗∗) The second equation gives x = y, sox = y = 1/1.001 and thus both variables have values that are just less than 1. A computer using two digits represents the system internally in this way (we will do this example in two-digit floating point arithmetic for clarity but inventing a similar one with eight or more digits is easy). (1.0 × 10 −3 ) · x +(1.0 × 10 0 ) · y = 1.0 × 10 0 (1.0 × 10 0 ) · x −(1.0 × 10 0 ) · y = 0.0 × 10 0 The row reduction step −1000ρ 1 + ρ 2 produces a second equation −1001y = −1000, which this computer rounds to two places as (−1.0 × 10 3 )y =−1.0 × 10 3 .
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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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12 Chapter One. Linear Systems 1.33
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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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Page 93 and 94: Chapter Two Vector Spaces The first
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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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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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