Linear Algebra

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60 Chapter One. <strong>Linear</strong> Systems(a)( )1 00 0(b)( )1 22 4(c)( )1 11 32.14 How many row equivalence classes are there?2.15 Can row equivalence classes contain different-sized matrices?2.16 How big are the row equivalence classes?(a) Show that the class of any zero matrix is finite.(b) Do any other classes contain only finitely many members?̌ 2.17 Give two reduced echelon form matrices that have their leading entries in thesame columns, but that are not row equivalent.̌ 2.18 Show that any two n×n nonsingular matrices are row equivalent. Are anytwo singular matrices row equivalent?̌ 2.19 Describe all of the row equivalence classes containing these.(a) 2 × 2 matrices (b) 2 × 3 matrices (c) 3 × 2 matrices(d) 3×3 matrices2.20 (a) Show that a vector β ⃗ 0 is a linear combination of members of the set{ β ⃗ 1, . . . , β ⃗ n} if and only if there is a linear relationship ⃗0 = c 0β0 ⃗ + · · · + c nβn⃗where c 0 is not zero. (Watch out for the β ⃗ 0 = ⃗0 case.)(b) Derive Lemma 2.5.̌ 2.21 Finish the proof of Lemma 2.5.(a) First illustrate the inductive step by showing that l 2 = k 2 .(b) Do the full inductive step: assume that c k is zero for 1 ≤ k < i − 1, anddeduce that c k+1 is also zero.(c) Find the contradiction.2.22 Finish the induction argument in Lemma 2.6.(a) State the inductive hypothesis, Also state what must be shown to follow fromthat hypothesis.(b) Check that the inductive hypothesis implies that in the relationship β r+1 =s r+1,1 δ 1 + s r+2,2 δ 2 + · · · + s r+1,m δ m the coefficients s r+1,1 , . . . , s r+1,r are eachzero.(c) Finish the inductive step by arguing, as in the base case, that l r+1 < k r+1and k r+1 < l r+1 are impossible.2.23 Why, in the proof of Theorem 2.7, do we bother to restrict to the nonzero rows?Why not just stick to the relationship that we began with, β i = c i,1 δ 1 +· · ·+c i,m δ m ,with m instead of r, and argue using it that the only nonzero coefficient is c i,i,which is 1?̌ 2.24 Three truck drivers went into a roadside cafe. One truck driver purchasedfour sandwiches, a cup of coffee, and ten doughnuts for $8.45. Another driverpurchased three sandwiches, a cup of coffee, and seven doughnuts for $6.30. Whatdid the third truck driver pay for a sandwich, a cup of coffee, and a doughnut?[Trono]2.25 The fact that Gaussian reduction disallows multiplication of a row by zero isneeded for the proof of uniqueness of reduced echelon form, or else every matrixwould be row equivalent to a matrix of all zeros. Where is it used?̌ 2.26 The <strong>Linear</strong> Combination Lemma says which equations can be gotten fromGaussian reduction from a given linear system.(1) Produce an equation not implied by this system.3x + 4y = 82x + y = 3
Section III. Reduced Echelon Form 61(2) Can any equation be derived from an inconsistent system?2.27 Extend the definition of row equivalence to linear systems. Under your definition,do equivalent systems have the same solution set? [Hoffman & Kunze]̌ 2.28 In this matrix ( 1 2) 33 0 31 4 5the first and second columns add to the third.(a) Show that remains true under any row operation.(b) Make a conjecture.(c) Prove that it holds.
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Linear AlgebraJim Hefferon( 13)( 21
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PrefaceThis book helps students to
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the material, but it’s only that
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Chapter Three: Maps Between Spaces
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Chapter OneLinear SystemsISolving L
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Section I. Solving Linear Systems 3
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Page 75 and 76: Topic: Input-Output Analysis 65For
Page 77 and 78: Topic: Input-Output Analysis 67(c)
Page 79 and 80: Topic: Accuracy of Computations 69(
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Page 83 and 84: Topic: Analyzing Networks 73differe
Page 85 and 86: Topic: Analyzing Networks 75Kirchof
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120 Chapter Two. Vector Spacesthat
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122 Chapter Two. Vector Spacesgreat
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126 Chapter Two. Vector SpacesNow t
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128 Chapter Two. Vector SpacesSo ou
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132 Chapter Two. Vector Spaces4.3 E
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134 Chapter Two. Vector SpacesProof
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138 Chapter Two. Vector Spaces(b) A
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140 Chapter Two. Vector SpacesTopic
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288 Chapter Four. DeterminantsIDefi
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290 Chapter Four. Determinantsalso
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292 Chapter Four. Determinantš 1.
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294 Chapter Four. DeterminantsProof
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296 Chapter Four. Determinants(a)
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298 Chapter Four. Determinantsand t
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300 Chapter Four. DeterminantsWe ar
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302 Chapter Four. Determinants3.8 E
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318 Chapter Four. Determinantš 1.
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320 Chapter Four. DeterminantsIIIOt
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322 Chapter Four. DeterminantsAlter
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324 Chapter Four. Determinants(a)(
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326 Chapter Four. Determinants(firs
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328 Chapter Four. DeterminantsTopic
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330 Chapter Four. Determinants(b) T
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Chapter FiveSimilarityWhile studyin
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Section I. Complex Vector Spaces 34
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Section II. Similarity 347IISimilar
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Section II. Similarity 3491.7 Consi
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Section II. Similarity 351Proof. Th
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Section II. Similarity 353(b) Is th
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Section II. Similarity 355The next
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Section II. Similarity 3573.13 Exam
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Section II. Similarity 3593.23 Find
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Section III. Nilpotence 361IIINilpo
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Section III. Nilpotence 363and this
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Section III. Nilpotence 365Proof. W
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Section III. Nilpotence 367The new
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Section III. Nilpotence 369Now, wit
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Section III. Nilpotence 371to the s
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Section III. Nilpotence 373( ) ( )
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Section IV. Jordan Form 375IVJordan
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Section IV. Jordan Form 377Using th
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Section IV. Jordan Form 379Equate t
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Section IV. Jordan Form 381̌ 1.17
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Section IV. Jordan Form 3832.3 Exam
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Section IV. Jordan Form 385their fi
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Section IV. Jordan Form 387Proof. S
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Section IV. Jordan Form 389Jordan f
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Section IV. Jordan Form 391the char
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Section IV. Jordan Form 393(b) The
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Topic: Method of Powers 395Topic: M
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Topic: Method of Powers 397(a)( )1
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Topic: Stable Populations 399Topic:
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Topic: Linear Recurrences 401Topic:
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Topic: Linear Recurrences 403And, i
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Topic: Linear Recurrences 405bee. O
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Topic: Linear Recurrences 4076 (Thi
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AppendixMathematics is made of argu
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A-3shows that Q holds whenever P do
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A-5more factors than any other’ w
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A-7IV.6 Sets, Functions, and Relati
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A-9We sometimes instead use the not
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A-11S 1 S 2S 0. . .S 3Thus, the fir
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Bibliography[Abbot] Edwin Abbott, F
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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composition, A-9self, 361computer a
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Kirchhoff’s Laws, 73Klein, F., 28
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at infinity, 337in projective plane
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vacuously true, A-2value, A-8Vander
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Linear Algebra

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