Linear Algebra, Theory And Applications, 2012a

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26 PRELIMINARIES Proof: If a equals a prime number, the prime factorization clearly exists. In particular the prime factorization exists for the prime number 2. Assume this theorem is true for all a ≤ n − 1. If n is a prime, then it has a prime factorization. On the other hand, if n is not a prime, then there exist two integers k and m such that n = km where each of k and m are less than n. Therefore, each of these is no larger than n − 1 and consequently, each has a prime factorization. Thus so does n. It remains to argue the prime factorization is unique except for order of the factors. Suppose n∏ m∏ p i = i=1 where the p i and q j are all prime, there is no way to reorder the q k such that m = n and p i = q i for all i, and n + m is the smallest positive integer such that this happens. Then by Theorem 1.9.7, p 1 |q j for some j. Since these are prime numbers this requires p 1 = q j . Reordering if necessary it can be assumed that q j = q 1 . Then dividing both sides by p 1 = q 1 , n−1 ∏ i=1 p i+1 = j=1 m−1 ∏ j=1 q j q j+1 . Since n + m was as small as possible for the theorem to fail, it follows that n − 1=m − 1 and the prime numbers, q 2 , ··· ,q m can be reordered in such a way that p k = q k for all k =2, ··· ,n. Hence p i = q i for all i because it was already argued that p 1 = q 1 , and this results in a contradiction. 1.10 Systems Of Equations Sometimes it is necessary to solve systems of equations. For example the problem could be to find x and y such that x + y =7and2x − y =8. (1.2) The set of ordered pairs, (x, y) which solve both equations is called the solution set. For example, you can see that (5, 2) = (x, y) is a solution to the above system. To solve this, note that the solution set does not change if any equation is replaced by a non zero multiple of itself. It also does not change if one equation is replaced by itself added to a multiple of the other equation. For example, x and y solve the above system if and only if x and y solve the system −3y=−6 { }} { x + y =7, 2x − y +(−2) (x + y) =8+(−2) (7). (1.3) The second equation was replaced by −2 times the first equation added to the second. Thus the solution is y =2, from −3y = −6 and now, knowing y =2, it follows from the other equation that x + 2 = 7 and so x =5. Why exactly does the replacement of one equation with a multiple of another added to it not change the solution set? The two equations of (1.2) are of the form E 1 = f 1 ,E 2 = f 2 (1.4) where E 1 and E 2 are expressions involving the variables. The claim is that if a is a number, then (1.4) has the same solution set as E 1 = f 1 ,E 2 + aE 1 = f 2 + af 1 . (1.5)
1.10. SYSTEMS OF EQUATIONS 27 Why is this? If (x, y) solves (1.4) then it solves the first equation in (1.5). Also, it satisfies aE 1 = af 1 and so, since it also solves E 2 = f 2 it must solve the second equation in (1.5). If (x, y) solves (1.5) then it solves the first equation of (1.4). Also aE 1 = af 1 and it is given that the second equation of (1.5) is verified. Therefore, E 2 = f 2 and it follows (x, y) isasolution of the second equation in (1.4). This shows the solutions to (1.4) and (1.5) are exactly the same which means they have the same solution set. Of course the same reasoning applies with no change if there are many more variables than two and many more equations than two. It is still the case that when one equation is replaced with a multiple of another one added to itself, the solution set of the whole system does not change. The other thing which does not change the solution set of a system of equations consists of listing the equations in a different order. Here is another example. Example 1.10.1 Find the solutions to the system, x +3y +6z =25 2x +7y +14z =58 2y +5z =19 (1.6) To solve this system replace the second equation by (−2) times the first equation added to the second. This yields. the system x +3y +6z =25 y +2z =8 2y +5z =19 (1.7) Now take (−2) times the second and add to the third. More precisely, replace the third equation with (−2) times the second added to the third. This yields the system x +3y +6z =25 y +2z =8 z =3 (1.8) At this point, you can tell what the solution is. This system has the same solution as the original system and in the above, z =3. Then using this in the second equation, it follows y + 6 = 8 and so y =2. Now using this in the top equation yields x + 6 + 18 = 25 and so x =1. This process is not really much different from what you have always done in solving a single equation. For example, suppose you wanted to solve 2x +5=3x − 6. You did the same thing to both sides of the equation thus preserving the solution set until you obtained an equation which was simple enough to give the answer. In this case, you would add −2x to both sides and then add 6 to both sides. This yields x =11. In (1.8) you could have continued as follows. Add (−2) times the bottom equation to the middle and then add (−6) times the bottom to the top. This yields x +3y =19 y =6 z =3 Now add (−3) times the second to the top. This yields x =1 y =6 z =3 ,
Page 1 and 2: Linear Algebra, Theory And Applicat
Page 3 and 4: Contents 1 Preliminaries 11 1.1 Set
Page 5 and 6: CONTENTS 5 8 Vector Spaces And Fiel
Page 7 and 8: CONTENTS 7 E The Fundamental Theore
Page 9 and 10: Preface This is a book on linear al
Page 11 and 12: Preliminaries 1.1 Sets And Set Nota
Page 13 and 14: 1.3. THE NUMBER LINE AND ALGEBRA OF
Page 15 and 16: 1.5. THE COMPLEX NUMBERS 15 Also fr
Page 17 and 18: 1.5. THE COMPLEX NUMBERS 17 and so
Page 19 and 20: 1.6. EXERCISES 19 Eventually, for r
Page 21 and 22: 1.8. WELL ORDERING AND ARCHIMEDEAN
Page 23 and 24: 1.9. DIVISION AND NUMBERS 23 Theore
Page 25: 1.9. DIVISION AND NUMBERS 25 Exampl
Page 29 and 30: 1.10. SYSTEMS OF EQUATIONS 29 The a
Page 31 and 32: 1.11. EXERCISES 31 It follows x =
Page 33 and 34: 1.14. EXERCISES 33 the existence of
Page 35 and 36: 1.15. THE INNER PRODUCT IN F N 35 s
Page 37 and 38: Matrices And Linear Transformations
Page 39 and 40: 2.1. MATRICES 39 Definition 2.1.2 M
Page 41 and 42: 2.1. MATRICES 41 is it possible to
Page 43 and 44: 2.1. MATRICES 43 a3× 3 matrix as d
Page 45 and 46: 2.1. MATRICES 45 1 2 3 4 Write the
Page 47 and 48: = ∑ k ( B T ) ik ( A T ) kj 2.1.
Page 49 and 50: 2.1. MATRICES 49 As described above
Page 51 and 52: 2.2. EXERCISES 51 Write the augment
Page 53 and 54: 2.3. LINEAR TRANSFORMATIONS 53 (e)
Page 55 and 56: 2.3. LINEAR TRANSFORMATIONS 55 Proo
Page 57 and 58: 2.4. SUBSPACES AND SPANS 57 Proof:
Page 59 and 60: 2.4. SUBSPACES AND SPANS 59 Proof 3
Page 61 and 62: 2.5. AN APPLICATION TO MATRICES 61
Page 63 and 64: 2.6. MATRICES AND CALCULUS 63 2.6.1
Page 65 and 66: 2.6. MATRICES AND CALCULUS 65 where
Page 67 and 68: 2.6. MATRICES AND CALCULUS 67 There
Page 69 and 70: 2.6. MATRICES AND CALCULUS 69 Using
Page 71 and 72: 2.7. EXERCISES 71 this will suffice
Page 73 and 74: 2.7. EXERCISES 73 22. If A is a lin
Page 75 and 76: 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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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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