Linear Algebra, 2020a

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422 Chapter Five. Similarity ̌ 3.31 Find the eigenvalues and associated eigenvectors of the differentiation operator d/dx: P 3 → P 3 . 3.32 Prove that the eigenvalues of a triangular matrix (upper or lower triangular) are the entries on the diagonal. ̌ 3.33 This matrix has distinct eigenvalues. ⎛ ⎝ 1 2 1 6 −1 0 −1 −2 −1 (a) Diagonalize it. (b) Find a basis with respect to which this matrix has that diagonal representation. (c) Draw the diagram. Find the matrices P and P −1 to effect the change of basis. ̌ 3.34 Find the formula for the characteristic polynomial of a 2×2 matrix. 3.35 Prove that the characteristic polynomial of a transformation is well-defined. 3.36 Prove or disprove: if all the eigenvalues of a matrix are 0 then it must be the zero matrix. ̌ 3.37 (a) Show that any non-⃗0 vector in any nontrivial vector space can be an eigenvector. That is, given a ⃗v ≠ ⃗0 from a nontrivial V, show that there is a transformation t: V → V having a scalar eigenvalue λ ∈ R such that ⃗v ∈ V λ . (b) What if we are given a scalar λ? Can any non-⃗0 member of any nontrivial vector space be an eigenvector associated with λ? ̌ 3.38 Suppose that t: V → V and T = Rep B,B (t). Prove that the eigenvectors of T associated with λ are the non-⃗0 vectors in the kernel of the map represented (with respect to the same bases) by T − λI. 3.39 Prove that if a,..., d are all integers and a + b = c + d then ( ) a b c d has integral eigenvalues, namely a + b and a − c. ̌ 3.40 Prove that if T is nonsingular and has eigenvalues λ 1 ,...,λ n then T −1 has eigenvalues 1/λ 1 ,...,1/λ n . Is the converse true? ̌ 3.41 Suppose that T is n×n and c, d are scalars. (a) Prove that if T has the eigenvalue λ with an associated eigenvector ⃗v then ⃗v is an eigenvector of cT + dI associated with eigenvalue cλ + d. (b) Prove that if T is diagonalizable then so is cT + dI. ̌ 3.42 Show that λ is an eigenvalue of T if and only if the map represented by T − λI is not an isomorphism. 3.43 [Strang 80] (a) Show that if λ is an eigenvalue of A then λ k is an eigenvalue of A k . (b) What is wrong with this proof generalizing that? “If λ is an eigenvalue of A and μ is an eigenvalue for B, then λμ is an eigenvalue for AB, for, if A⃗x = λ⃗x and B⃗x = μ⃗x then AB⃗x = Aμ⃗x = μA⃗x = μλ⃗x”? 3.44 Do matrix equivalent matrices have the same eigenvalues? 3.45 Show that a square matrix with real entries and an odd number of rows has at least one real eigenvalue. ⎞ ⎠
Section II. Similarity 423 3.46 Diagonalize. ⎛ −1 2 ⎞ 2 ⎝ 2 2 2 ⎠ −3 −6 −6 3.47 Suppose that P is a nonsingular n × n matrix. Show that the similarity transformation map t P : M n×n → M n×n sending T ↦→ PTP −1 is an isomorphism. ? 3.48 [Math. Mag., Nov. 1967] Show that if A is an n square matrix and each row (column) sums to c then c is a characteristic root of A. (“Characteristic root” is a synonym for eigenvalue.)
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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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Page 475 and 476: Topic: Method of Powers 465 ⃗v T
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Page 479 and 480: Topic: Stable Populations 469 Now i
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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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