Linear Algebra, Theory And Applications, 2012a

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492 ANSWERS TO SELECTED EXERCISES G.11 Exercises 6.6 1 The maximum is 7 and it occurs when x 1 =7,x 2 = 0,x 3 =0,x 4 =3,x 5 =5,x 6 =0. 2 Maximize and minimize the following if possible. All variables are nonnegative. (a) The minimum is −7 and it happens when x 1 = 0,x 2 =7/2,x 3 =0. (b) The maximum is 7 and it occurs when x 1 = 7,x 2 =0,x 3 =0. (c) The maximum is 14 and it happens when x 1 = 7,x 2 = x 3 =0. (d) The minimum is 0 when x 1 = x 2 =0,x 3 =1. 4 Find a solution to the following inequalities for x, y ≥ 0 if it is possible to do so. If it is not possible, prove it is not possible. (a) There is no solution to these inequalities with x 1 ,x 2 ≥ 0. (b) A solution is x 1 =8/5,x 2 = x 3 =0. (c) There will be no solution to these inequalities for which all the variables are nonnegative. (d) There is a solution when x 2 =2,x 3 =0,x 1 = 0. (e) There is no solution to this system of inequalities because the minimum value of x 7 is not 0. G.12 Exercises 7.3 1 Because the vectors which result are not parallel to the vector you begin with. 3 λ → λ −1 and λ → λ m . 5 Letx be the eigenvector. Then A m x = λ m x,A m x = Ax = λx and so λ m = λ Hence if λ ≠0, then and so |λ| =1. λ m−1 =1 7 9 11 ⎛ ⎝ ⎧ ⎨ −1 −1 7 −1 0 4 −1 −1 5 ⎫ ⎬ ⎞ ⎠, eigenvectors: ⎫ ⎬ ⎧ 3 ⎨ 2 1 ⎩ ⎭ ↔ 1, 1 ↔ 2. This is a defective matrix. ⎩ ⎭ 1 1 ⎛ ⎞ −7 −12 30 ⎝ −3 −7 15 ⎠, eigenvectors: −3 −6 14 ⎧⎛ ⎨ ⎝ −2 ⎞ ⎛ 1 ⎠ , ⎝ 5 ⎞⎫ ⎧⎛ ⎬ ⎨ 0 ⎠ ⎩ ⎭ ↔−1, ⎝ 2 ⎞⎫ ⎬ 1 ⎠ ⎩ ⎭ ↔ 2 0 1 1 This matrix is not defective because, even though λ = 1 is a repeated eigenvalue, it has a 2 dimensional eigenspace. ⎛ ⎝ 3 −2 −1 0 5 1 0 2 4 ⎞ ⎠, eigenvectors: ⎧⎛ ⎞ ⎛ ⎞⎫ ⎧⎛ ⎨ 1 0 ⎬ ⎨ ⎝ 0 ⎠ , ⎝ − 1 ⎠ ⎩ 2 ⎭ ↔ 3, ⎝ ⎩ 0 1 This matrix is not defective. ⎛ 13 ⎝ 5 2 −5 ⎞ 12 3 −10 ⎠, eigenvectors: 12 4 −11 ⎧⎛ ⎨ ⎝ − ⎞ ⎛ ⎞⎫ 1 5 3 6 ⎬ 1 ⎠ , ⎝ 0 ⎠ ⎩ ⎭ ↔−1 0 1 15 17 −1 1 1 ⎞⎫ ⎬ ⎠ ⎭ ↔ 6 This matrix is defective. In this case, there is only one eigenvalue, −1 of multiplicity 3 but the dimension of the eigenspace is only 2. ⎛ ⎝ 1 26 −17 4 −4 4 −9 −18 9 ⎧⎛ ⎞⎫ ⎧⎛ ⎨ − 1 3 ⎬ ⎨ ⎝ 2 ⎠ ⎩ 3 ⎭ ↔ 0, ⎝ ⎩ 1 ⎧⎛ ⎨ ⎝ −1 ⎞⎫ ⎬ 0 ⎠ ⎩ ⎭ ↔ 18 1 ⎞ ⎠, eigenvectors: −2 1 0 ⎞⎫ ⎬ ⎠ ⎭ ↔−12, ⎛ ⎝ −2 1 2 ⎞ ⎧⎛ ⎨ −11 −2 9 ⎠, eigenvectors: ⎝ ⎩ −8 0 7 This is defective. 3 4 1 4 1 ⎞⎫ ⎬ ⎠ ⎭ ↔ 1
G.13. EXERCISES 493 19 ⎛ ⎝ 4 −2 −2 0 2 −2 2 0 2 ⎞ ⎠, eigenvectors: ⎧⎛ ⎨ ⎝ 1 ⎞⎫ ⎧⎛ ⎬ ⎨ −1 ⎠ ⎩ ⎭ ↔ 4, ⎝ −i ⎞⎫ ⎬ −i ⎠ ⎩ ⎭ ↔ 2 − 2i, 1 1 ⎧⎛ ⎨ ⎝ i ⎞⎫ ⎬ i ⎠ ⎩ ⎭ ↔ 2+2i 1 ⎛ ⎞ 4 −2 −2 21 ⎝ 0 2 −2 ⎠, eigenvectors: 2 0 2 ⎧⎛ ⎨ ⎝ 1 ⎞⎫ ⎧⎛ ⎬ ⎨ −1 ⎠ ⎩ ⎭ ↔ 4, ⎝ −i ⎞⎫ ⎬ −i ⎠ ⎩ ⎭ ↔ 2 − 2i, 1 1 ⎧⎛ ⎞⎫ ⎨ i ⎬ ⎝ i ⎠ ⎩ ⎭ ↔ 2+2i 1 ⎛ 23 ⎝ 1 1 −6 ⎞ 7 −5 −6 ⎠, eigenvectors: −1 7 2 ⎧⎛ ⎞⎫ ⎧⎛ ⎞⎫ ⎨ 1 ⎬ ⎨ −i ⎬ ⎝ −1 ⎠ ⎩ ⎭ ↔−6, ⎝ −i ⎠ ⎩ ⎭ ↔ 2 − 6i, 1 1 ⎧⎛ ⎨ ⎝ i ⎞⎫ ⎬ i ⎠ ⎩ ⎭ ↔ 2+6i 1 This is not defective. 25 First consider the eigenvalue λ =1. Then you have ax 2 = 0,bx 3 = 0. If neither a nor b = 0 then λ = 1 would be a defective eigenvalue and the matrix would be defective. If a =0, then the dimension of the eigenspace is clearly 2 and so the matrix would be nondefective. If b = 0 but a ≠0, then you would have a defective matrix because the eigenspace would have dimension less than 2. If c ≠0, then the matrix is defective. If c = 0 and a =0, then it is non defective. Basically, if a, c ≠0, then the matrix is defective. 27 A (x + iy) =(a + ib)(x + iy) . Now just take complex conjugates of both sides. 29 Let A be skew symmetric. Then if x is an eigenvector for λ, λx T ¯x = x T A T ¯x = −x T A¯x = −x T ¯x¯λ and so λ = −¯λ. Thus a + ib = − (a − ib) andso a =0. 31 This follows from the observation that if Ax = λx, then Ax = λx ⎛⎛ 33 ⎝⎝ ⎞ ⎞ ⎛⎛ −1 ⎠ , 1⎠ , ⎝⎝ ⎞ ⎞ ⎛⎛ 1 2 1 ⎠ 2 , 1 ⎠ 2 , ⎝⎝ ⎞ ⎞ 1 ⎠ , 1 ⎠ 3 1 1 0 ⎛ ⎞ −1 35 ⎝ 1 ⎠ (a cos (t)+b sin (t)) , ⎛ 1 ⎞ ⎝ 0 −1 ⎠ ( c sin (√ 2t ) + d cos (√ 2t )) , ⎛ 1 ⎞ ⎝ 2 1 ⎠ (e cos (2t)+f sin (2t))where a, b, c, d, e, f are 1 scalars. G.13 Exercises 7.10 1 To get it, you must be able to get the eigenvalues and this is typically not possible. ( ) ( )( ) 0 −1 0 −1 2 0 4 = 2 0 1 0 0 1 ( )( ) 2 0 0 −1 A 1 = 0 1 1 0 ( ) 0 −2 = 1 0 ( ) ( )( ) 0 −2 0 −1 1 0 = 1 0 1 0 0 2 ( )( ) ( ) 1 0 0 −1 0 −1 A 2 = = . Now 0 2 1 0 2 0 it is back to where you started. Thus the algorithm ( merely ) bounces ( between ) the two matrices 0 −1 0 −2 and and so it can’t possibly 2 0 1 0 converge. 15 B (1 + 2i, 6) ,B(i, 3) ,B(7, 11) 19 Gerschgorin’s theorem shows that there are no zero eigenvalues and so the matrix is invertible. 21 6x ′2 +12y ′2 +18z ′2 . 23 (x ′ ) 2 + 1 3√ 3x ′ − 2(y ′ ) 2 − 1 2√ 2y ′ − 2(z ′ ) 2 − 1 6√ 6z ′
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Linear Algebra, Theory And Applicat
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Contents 1 Preliminaries 11 1.1 Set
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CONTENTS 5 8 Vector Spaces And Fiel
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CONTENTS 7 E The Fundamental Theore
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Preface This is a book on linear al
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Preliminaries 1.1 Sets And Set Nota
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1.3. THE NUMBER LINE AND ALGEBRA OF
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1.5. THE COMPLEX NUMBERS 15 Also fr
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1.5. THE COMPLEX NUMBERS 17 and so
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1.6. EXERCISES 19 Eventually, for r
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1.8. WELL ORDERING AND ARCHIMEDEAN
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1.9. DIVISION AND NUMBERS 23 Theore
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1.9. DIVISION AND NUMBERS 25 Exampl
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1.10. SYSTEMS OF EQUATIONS 27 Why i
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1.10. SYSTEMS OF EQUATIONS 29 The a
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1.11. EXERCISES 31 It follows x =
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1.14. EXERCISES 33 the existence of
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1.15. THE INNER PRODUCT IN F N 35 s
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Matrices And Linear Transformations
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2.1. MATRICES 39 Definition 2.1.2 M
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2.1. MATRICES 41 is it possible to
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2.1. MATRICES 43 a3× 3 matrix as d
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2.1. MATRICES 45 1 2 3 4 Write the
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= ∑ k ( B T ) ik ( A T ) kj 2.1.
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2.1. MATRICES 49 As described above
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2.2. EXERCISES 51 Write the augment
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2.3. LINEAR TRANSFORMATIONS 53 (e)
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2.3. LINEAR TRANSFORMATIONS 55 Proo
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2.4. SUBSPACES AND SPANS 57 Proof:
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2.4. SUBSPACES AND SPANS 59 Proof 3
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2.5. AN APPLICATION TO MATRICES 61
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2.6. MATRICES AND CALCULUS 63 2.6.1
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2.6. MATRICES AND CALCULUS 65 where
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2.6. MATRICES AND CALCULUS 67 There
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2.6. MATRICES AND CALCULUS 69 Using
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2.7. EXERCISES 71 this will suffice
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2.7. EXERCISES 73 22. If A is a lin
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2.7. EXERCISES 75 35. ↑Suppose yo
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Determinants 3.1 Basic Techniques A
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3.1. BASIC TECHNIQUES AND PROPERTIE
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3.2. EXERCISES 81 First note det (A
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3.3. THE MATHEMATICAL THEORY OF DET
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3.4. THE CAYLEY HAMILTON THEOREM 97
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.5. BLOCK MULTIPLICATION OF MATRIC
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3.6. EXERCISES 103 derivatives so i
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Row Operations 4.1 Elementary Matri
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4.1. ELEMENTARY MATRICES 107 This h
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4.1. ELEMENTARY MATRICES 109 Now co
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4.2. THE RANK OF A MATRIX 111 So ho
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.3. THE ROW REDUCED ECHELON FORM 1
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4.5. FREDHOLM ALTERNATIVE 117 Proof
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4.6. EXERCISES 119 5. ↑Consider t
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4.6. EXERCISES 121 18. Suppose A is
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Some Factorizations 5.1 LU Factoriz
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5.3. SOLVING LINEAR SYSTEMS USING A
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5.5. JUSTIFICATION FOR THE MULTIPLI
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5.6. EXISTENCE FOR THE PLU FACTORIZ
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5.7. THE QR FACTORIZATION 131 so it
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5.8. EXERCISES 133 Theorem 5.7.5 Le
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Linear Programming 6.1 Simple Geome
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6.2. THE SIMPLEX TABLEAU 137 set x
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6.2. THE SIMPLEX TABLEAU 139 lution
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6.3. THE SIMPLEX ALGORITHM 141 Let
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6.3. THE SIMPLEX ALGORITHM 143 the
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6.3. THE SIMPLEX ALGORITHM 145 by 2
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6.3. THE SIMPLEX ALGORITHM 147 I ne
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6.3. THE SIMPLEX ALGORITHM 149 Of c
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6.4. FINDING A BASIC FEASIBLE SOLUT
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6.5. DUALITY 153 Now recall that to
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6.5. DUALITY 155 and there are stil
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Spectral Theory Spectral Theory ref
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7.1. EIGENVALUES AND EIGENVECTORS O
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7.2. SOME APPLICATIONS OF EIGENVALU
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7.3. EXERCISES 167 Therefore, the e
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7.3. EXERCISES 169 22. Find the com
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7.3. EXERCISES 171 38. Let M be an
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7.4. SCHUR’S THEOREM 173 7.4 Schu
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7.4. SCHUR’S THEOREM 175 Proof: L
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7.4. SCHUR’S THEOREM 177 and furt
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7.4. SCHUR’S THEOREM 179 Proof: F
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7.6. QUADRATIC FORMS 181 7.6 Quadra
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7.7. SECOND DERIVATIVE TEST 183 Cor
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7.7. SECOND DERIVATIVE TEST 185 The
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7.9. ADVANCED THEOREMS 187 for find
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7.9. ADVANCED THEOREMS 189 Proof: D
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7.10. EXERCISES 191 Show that the a
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7.10. EXERCISES 193 27. Let f (x, y
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7.10. EXERCISES 195 40. In the abov
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7.10. EXERCISES 197 44. Show that i
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Vector Spaces And Fields 8.1 Vector
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8.2. SUBSPACES AND BASES 201 8.2.2
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8.2. SUBSPACES AND BASES 203 Defini
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8.3. LOTS OF FIELDS 205 such that 0
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8.3. LOTS OF FIELDS 207 and this is
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8.3. LOTS OF FIELDS 209 Lemma 8.3.9
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8.3. LOTS OF FIELDS 211 Definition
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8.3. LOTS OF FIELDS 213 Then, since
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8.3. LOTS OF FIELDS 215 Proposition
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8.3. LOTS OF FIELDS 217 where deg r
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8.4. EXERCISES 219 8.3.4 The Lindem
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8.4. EXERCISES 221 18. If you have
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8.4. EXERCISES 223 first is not har
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Linear Transformations 9.1 Matrix M
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9.3. THE MATRIX OF A LINEAR TRANSFO
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9.5. EXERCISES 243 9. ↑In the sit
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Linear Transformations Canonical Fo
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10.1. A THEOREM OF SYLVESTER, DIREC
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10.2. DIRECT SUMS, BLOCK DIAGONAL M
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10.3. CYCLIC SETS 251 while ⎛ ⎞
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10.3. CYCLIC SETS 253 Since β x is
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10.4. NILPOTENT TRANSFORMATIONS 255
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10.5. THE JORDAN CANONICAL FORM 257
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10.6. EXERCISES 263 8. Let ⎛ A =
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10.6. EXERCISES 265 Now use this in
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10.7. THE RATIONAL CANONICAL FORM 2
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10.8. UNIQUENESS 269 Thus the matri
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10.8. UNIQUENESS 271 The characteri
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10.9. EXERCISES 273 Then doing the
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Markov Chains And Migration Process
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11.1. REGULAR MARKOV MATRICES 277 O
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11.2. MIGRATION MATRICES 279 11.2 M
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11.3. MARKOV CHAINS 281 After many
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11.3. MARKOV CHAINS 283 and the fac
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11.4. EXERCISES 285 5. Show that wh
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Inner Product Spaces 12.1 General T
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12.2. THE GRAM SCHMIDT PROCESS 289
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12.2. THE GRAM SCHMIDT PROCESS 291
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12.3. RIESZ REPRESENTATION THEOREM
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12.4. THE TENSOR PRODUCT OF TWO VEC
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12.5. LEAST SQUARES 297 Lemma 12.5.
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12.7. EXERCISES 299 4. Let ||x||
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12.7. EXERCISES 301 (√ √ ) 2 C
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12.8. THE DETERMINANT AND VOLUME 30
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12.8. THE DETERMINANT AND VOLUME 30
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Self Adjoint Operators 13.1 Simulta
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13.1. SIMULTANEOUS DIAGONALIZATION
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13.2. SCHUR’S THEOREM 311 Proof:
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.3. SPECTRAL THEORY OF SELF ADJOI
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13.4. POSITIVE AND NEGATIVE LINEAR
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13.5. FRACTIONAL POWERS 319 Proof:
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13.5. FRACTIONAL POWERS 321 Proof:
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13.6. POLAR DECOMPOSITIONS 323 Proo
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13.7. AN APPLICATION TO STATISTICS
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13.8. THE SINGULAR VALUE DECOMPOSIT
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13.9. APPROXIMATION IN THE FROBENIU
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13.10. LEAST SQUARES AND SINGULAR V
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13.11. THE MOORE PENROSE INVERSE 33
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13.12. EXERCISES 335 8. Prove the C
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Norms For Finite Dimensional Vector
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339 This proves the first half of t
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341 Next consider the claim that ||
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14.1. THE P NORMS 343 14.1 The p No
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14.2. THE CONDITION NUMBER 345 Thus
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14.2. THE CONDITION NUMBER 347 Sinc
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14.3. THE SPECTRAL RADIUS 349 ⎛ =
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14.4. SERIES AND SEQUENCES OF LINEA
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14.4. SERIES AND SEQUENCES OF LINEA
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14.5. ITERATIVE METHODS FOR LINEAR
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14.6. THEORY OF CONVERGENCE 361 Lem
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14.7. EXERCISES 363 Proof: From Lem
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14.7. EXERCISES 365 11. Suppose ρ
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14.7. EXERCISES 367 Next take e tλ
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14.7. EXERCISES 369 24. Suppose A i
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Numerical Methods For Finding Eigen
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15.1. THE POWER METHOD FOR EIGENVAL
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15.2. THE QR ALGORITHM 387 each a v
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15.2. THE QR ALGORITHM 389 Proof: L
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15.2. THE QR ALGORITHM 391 where
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15.2. THE QR ALGORITHM 393 Corollar
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15.2. THE QR ALGORITHM 395 below th
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15.2. THE QR ALGORITHM 397 where R
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15.2. THE QR ALGORITHM 399 A matrix
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15.3. EXERCISES 401 where A k = Q k
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Positive Matrices Earlier theorems
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405 |μ| ≤λ 0 . But also, the fa
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407 which says x 0 −x 0 v T x 0 =
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409 must have the same argument for
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Functions Of Matrices The existence
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413 There is no convergence problem
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415 This gives us a nice definition
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Applications To Differential Equati
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C.3. LOCAL SOLUTIONS 419 C.3 Local
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C.4. FIRST ORDER LINEAR SYSTEMS 421
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.5. GEOMETRIC THEORY OF AUTONOMOUS
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C.6. GENERAL GEOMETRIC THEORY 433 T
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C.7. THESTABLEMANIFOLD 435 Lemma C.
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C.7. THESTABLEMANIFOLD 437 ≤ e γ
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Compactness And Completeness D.0.1
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Page 443 and 444: The Fundamental Theorem Of Algebra
Page 445 and 446: Fields And Field Extensions F.1 The
Page 447 and 448: F.2. THE FUNDAMENTAL THEOREM OF ALG
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Page 451 and 452: F.3. TRANSCENDENTAL NUMBERS 451 F.3
Page 453 and 454: F.3. TRANSCENDENTAL NUMBERS 453 It
Page 455 and 456: F.3. TRANSCENDENTAL NUMBERS 455 Not
Page 457 and 458: F.3. TRANSCENDENTAL NUMBERS 457 Thi
Page 459 and 460: F.4. MORE ON ALGEBRAIC FIELD EXTENS
Page 465 and 466: F.5. THE GALOIS GROUP 465 a rationa
Page 467 and 468: F.5. THE GALOIS GROUP 467 Now let
Page 469 and 470: F.6. NORMAL SUBGROUPS 469 If H is a
Page 471 and 472: F.8. CONDITIONS FOR SEPARABILITY 47
Page 473 and 474: F.8. CONDITIONS FOR SEPARABILITY 47
Page 475 and 476: F.9. PERMUTATIONS 475 of f (x) ands
Page 477 and 478: F.9. PERMUTATIONS 477 smallest k
Page 479 and 480: F.10. SOLVABLE GROUPS 479 Then i 1
Page 481 and 482: F.10. SOLVABLE GROUPS 481 Thus ever
Page 483 and 484: F.11. SOLVABILITY BY RADICALS 483 A
Page 485 and 486: Bibliography [1] Apostol T., Calcul
Page 487 and 488: Answers To Selected Exercises G.1 E
Page 489 and 490: G.7. EXERCISES 489 15 Find the matr
Page 491: G.10. EXERCISES 491 11 It follows f
Page 495 and 496: G.19. EXERCISES 495 8 λ 3 − λ 2
Page 497 and 498: G.23. EXERCISES 497 ⎧⎛ ⎞⎫
Page 499 and 500: INDEX 499 defective, 162 DeMoivre i
Page 501 and 502: INDEX 501 rotation, 235 linear tran
Page 503: INDEX 503 scalars, 18, 32, 37 Schur
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