Linear Algebra, Theory And Applications, 2012a

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134 SOME FACTORIZATIONS 9. Find a QR factorization for the matrix ⎛ ⎝ 1 2 1 0 3 0 1 1 1 0 2 1 10. If you had a QR factorization, A = QR, describe how you could use it to solve the equation Ax = b. 11. If Q is an orthogonal matrix, show the columns are an orthonormal set. That is show that for Q = ( ) q 1 ··· q n it follows that q i · q j = δ ij . Also show that any orthonormal set of vectors is linearly independent. 12. Show you can’t expect uniqueness for QR factorizations. Consider ⎛ ⎝ 0 0 0 0 0 1 ⎞ ⎠ 0 0 1 and verify this equals and also ⎛ ⎝ ⎞ ⎛ √ 0 1 0 2 0 1 √ 2√ 2 ⎠ √ 2 0 − 1 2 2 1 2 1 2 ⎛ ⎝ 1 0 0 0 1 0 0 0 1 ⎞ ⎛ ⎠ ⎝ ⎞ ⎠ ⎝ 0 0 √ 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 Using Definition 5.7.4, can it be concluded that if A is an invertible matrix it will follow there is only one QR factorization? ⎞ ⎠ . 13. Suppose {a 1 , ··· , a n } are linearly independent vectors in R n and let A = ( a 1 ··· a n ) ⎞ ⎠ Form a QR factorization for A. ⎛ ( ) ( ) a1 ··· a n = q1 ··· q n ⎜ ⎝ ⎞ r 11 r 12 ··· r 1n 0 r 22 ··· r 2n . . .. ⎟ ⎠ 0 0 ··· r nn Show that for each k ≤ n, span (a 1 , ··· , a k ) = span (q 1 , ··· , q k ) Prove that every subspace of R n has an orthonormal basis. The procedure just described is similar to the Gram Schmidt procedure which will be presented later. 14. Suppose Q n R n converges to an orthogonal matrix Q where Q n is orthogonal and R n is upper triangular having all positive entries on the diagonal. Show that then Q n converges to Q and R n converges to the identity.
<strong>Linear</strong> Programming 6.1 Simple Geometric Considerations One of the most important uses of row operations is in solving linear program problems which involve maximizing a linear function subject to inequality constraints determined from linear equations. Here is an example. A certain hamburger store has 9000 hamburger patties to use in one week and a limitless supply of special sauce, lettuce, tomatoes, onions, and buns. They sell two types of hamburgers, the big stack and the basic burger. It has also been determined that the employees cannot prepare more than 9000 of either type in one week. The big stack, popular with the teenagers from the local high school, involves two patties, lots of delicious sauce, condiments galore, and a divider between the two patties. The basic burger, very popular with children, involves only one patty and some pickles and ketchup. Demand for the basic burger is twice what it is for the big stack. What is the maximum number of hamburgers which could be sold in one week given the above limitations? Let x be the number of basic burgers and y the number of big stacks which could be sold in a week. Thus it is desired to maximize z = x + y subject to the above constraints. The total number of patties is 9000 and so the number of patty used is x+2y. This number must satisfy x +2y ≤ 9000 because there are only 9000 patty available. Because of the limitation on the number the employees can prepare and the demand, it follows 2x + y ≤ 9000. You never sell a negative number of hamburgers and so x, y ≥ 0. In simpler terms the problem reduces to maximizing z = x + y subject to the two constraints, x +2y ≤ 9000 and 2x + y ≤ 9000. This problem is pretty easy to solve geometrically. Consider the following picture in which R labels the region described by the above inequalities and the line z = x+y is shown for a particular value of z. x + y = z 2x + y =4 R x +2y =4 As you make z larger this line moves away from the origin, always having the same slope 135
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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
Page 83 and 84: 3.3. THE MATHEMATICAL THEORY OF DET
Page 97 and 98: 3.4. THE CAYLEY HAMILTON THEOREM 97
Page 99 and 100: 3.5. BLOCK MULTIPLICATION OF MATRIC
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Page 103 and 104: 3.6. EXERCISES 103 derivatives so i
Page 105 and 106: Row Operations 4.1 Elementary Matri
Page 107 and 108: 4.1. ELEMENTARY MATRICES 107 This h
Page 109 and 110: 4.1. ELEMENTARY MATRICES 109 Now co
Page 111 and 112: 4.2. THE RANK OF A MATRIX 111 So ho
Page 113 and 114: 4.3. THE ROW REDUCED ECHELON FORM 1
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Page 117 and 118: 4.5. FREDHOLM ALTERNATIVE 117 Proof
Page 119 and 120: 4.6. EXERCISES 119 5. ↑Consider t
Page 121 and 122: 4.6. EXERCISES 121 18. Suppose A is
Page 123 and 124: Some Factorizations 5.1 LU Factoriz
Page 125 and 126: 5.3. SOLVING LINEAR SYSTEMS USING A
Page 127 and 128: 5.5. JUSTIFICATION FOR THE MULTIPLI
Page 129 and 130: 5.6. EXISTENCE FOR THE PLU FACTORIZ
Page 131 and 132: 5.7. THE QR FACTORIZATION 131 so it
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Page 137 and 138: 6.2. THE SIMPLEX TABLEAU 137 set x
Page 139 and 140: 6.2. THE SIMPLEX TABLEAU 139 lution
Page 141 and 142: 6.3. THE SIMPLEX ALGORITHM 141 Let
Page 143 and 144: 6.3. THE SIMPLEX ALGORITHM 143 the
Page 145 and 146: 6.3. THE SIMPLEX ALGORITHM 145 by 2
Page 147 and 148: 6.3. THE SIMPLEX ALGORITHM 147 I ne
Page 149 and 150: 6.3. THE SIMPLEX ALGORITHM 149 Of c
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Page 153 and 154: 6.5. DUALITY 153 Now recall that to
Page 155 and 156: 6.5. DUALITY 155 and there are stil
Page 157 and 158: Spectral Theory Spectral Theory ref
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Page 167 and 168: 7.3. EXERCISES 167 Therefore, the e
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Page 171 and 172: 7.3. EXERCISES 171 38. Let M be an
Page 173 and 174: 7.4. SCHUR’S THEOREM 173 7.4 Schu
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Page 177 and 178: 7.4. SCHUR’S THEOREM 177 and furt
Page 179 and 180: 7.4. SCHUR’S THEOREM 179 Proof: F
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Page 183 and 184: 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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9.4. EIGENVALUES AND EIGENVECTORS O
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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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441 |a k − a l | < ε 2 .Thenifk>
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The Fundamental Theorem Of Algebra
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Fields And Field Extensions F.1 The
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.2. THE FUNDAMENTAL THEOREM OF ALG
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F.3. TRANSCENDENTAL NUMBERS 451 F.3
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F.3. TRANSCENDENTAL NUMBERS 453 It
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F.3. TRANSCENDENTAL NUMBERS 455 Not
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F.3. TRANSCENDENTAL NUMBERS 457 Thi
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F.4. MORE ON ALGEBRAIC FIELD EXTENS
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F.5. THE GALOIS GROUP 465 a rationa
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F.5. THE GALOIS GROUP 467 Now let
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F.6. NORMAL SUBGROUPS 469 If H is a
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.8. CONDITIONS FOR SEPARABILITY 47
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F.9. PERMUTATIONS 475 of f (x) ands
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F.9. PERMUTATIONS 477 smallest k
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F.10. SOLVABLE GROUPS 479 Then i 1
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F.10. SOLVABLE GROUPS 481 Thus ever
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F.11. SOLVABILITY BY RADICALS 483 A
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Bibliography [1] Apostol T., Calcul
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Answers To Selected Exercises G.1 E
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G.7. EXERCISES 489 15 Find the matr
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G.10. EXERCISES 491 11 It follows f
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G.13. EXERCISES 493 19 ⎛ ⎝ 4
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G.19. EXERCISES 495 8 λ 3 − λ 2
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G.23. EXERCISES 497 ⎧⎛ ⎞⎫
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INDEX 499 defective, 162 DeMoivre i
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INDEX 501 rotation, 235 linear tran
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INDEX 503 scalars, 18, 32, 37 Schur
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