Linear Algebra, Theory And Applications, 2012a

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476 FIELDS AND FIELD EXTENSIONS Proof: This is obvious if n = 1 or 2. Suppose then that it is true for sets of n−1elements. Take two permutations of {a 1 , ··· ,a n } ,P 1 ,P 2 . To get from P 1 to P 2 using switches, first make a switch to obtain the last element in the list coinciding with the last element of P 2 . By induction, there are switches which will arrange the first n − 1 to the right order. It is customary to consider permutations in terms of the set I n ≡{1, ··· ,n} to be more specific. Then one can think of a given permutation as a mapping σ from this set I n to itself which is one to one and onto. In fact, σ (i) ≡ j where j is in the i th position. Often people write such a σ in the following form ( ) 1 2 ··· n (6.21) i 1 i 2 ··· i n An easy way to understand the above permutation is through the use of matrix multiplication by permutation matrices. The above vector (i 1 , ··· ,i n ) T is obtained by ⎛ ( ) ei1 e i2 ··· e in ⎜ ⎝ 1 2 . n ⎞ ⎟ ⎠ (6.22) This can be seen right away from looking at a simple example or by using the definition of matrix multiplication directly. Definition F.9.2 The sign of the permutation (6.21) is defined as the determinant of the above matrix in (6.22). In other words, the sign of the permutation ( ) 1 2 ··· n i 1 i 2 ··· i n equals sgn (i 1 , ··· ,i n ) defined earlier in Lemma 3.3.1. Note that from the fact that the determinant is well defined and its properties, the sign of a permutation is 1 if and only if the permutation is produced by an even number of switches and that the number of switches used to produce a given permutation must be either even or odd. Of course a switch is a permutation itself and this is called a transposition. Note also that all these matrices are orthogonal matrices so to take the inverse, it suffices to take a transpose, the inverse also being a permutation matrix. The resulting group consisting of the permutations of I n is called S n .Animportantidea is the notion of a cycle. Let σ be a permutation, a one to one and onto function defined on I n . A cycle is of the form ( k, σ (k) ,σ 2 (k) ,σ 3 (k) , ··· ,σ m−1 (k) ) ,σ m (k) =k. The last condition must hold for some m because I n is finite. Then a cycle can be considered as a permutation as follows. Let (i 1 ,i 2 , ··· ,i m ) be a cycle. Then define σ by σ (i 1 )= i 2 ,σ(i 2 )=i 3 , ··· ,σ(i m )=i 1 ,andifk/∈{i 1 ,i 2 , ··· ,i m } , then σ (k) =k. Note that if you have two cycles, (i 1 ,i 2 , ··· ,i m ) , (j 1 ,j 2 , ··· ,j m ) which are disjoint in the sense that {i 1 ,i 2 , ··· ,i m }∩{j 1 ,j 2 , ··· ,j m } = ∅, then they commute. It is then clear that every permutation can be represented in a unique way by disjoint cycles. Start with 1 and form the cycle determined by 1. Then start with the
F.9. PERMUTATIONS 477 smallest k ∈ I n which was not included and begin a cycle starting with this. Continue this way. Use the convention that (k) is just the identity. This representation is unique up to order of the cycles which does not matter because they commute. Note that a transposition canbewrittenas(a, b). A cycle can be written as a product of non disjoint transpositions. (i 1 ,i 2 , ··· ,i m )=(i m−1 ,i m ) ···(i 2 ,i m )(i 1 ,i m ) Thus if m is odd, the permutation has sign 1 and if m is even, the permutation has sign −1. Also, it is clear that the inverse of the above permutation is (i 1 ,i 2 , ··· ,i m ) −1 = (i m , ··· ,i 2 ,i 1 ) . Definition F.9.3 A n is the subgroup of S n such that for σ ∈ A n , σ is the product of an even number of transpositions. It is called the alternating group. The following important result is useful in describing A n . Proposition F.9.4 Let n ≥ 3. Then every permutation in A n is the product of 3 cycles and the identity. Proof: In case n =3, you can list all of the permutations in A n ( ) ( ) ( ) 1 2 3 1 2 3 1 2 3 , , 1 2 3 2 3 1 3 1 2 In terms of cycles, these are (1, 2, 3) , (1, 3, 2) You can easily check that they are inverses of each other. Now suppose n ≥ 4. The permutations in A n are defined as the product of an even number of transpositions. There are two cases. The first case is where you have two transpositions which share a number, (a, c)(c, b) =(a, c, b) Thus when they share a number, the product is just a 3 cycle. Next suppose you have the product of two transpositions which are disjoint. This can happen because n ≥ 4. First note that (a, b) =(c, b)(b, a, c) =(c, b, a)(c, a) Therefore, (a, b)(c, d) = (c, b, a)(c, a)(a, d)(d, c, a) = (c, b, a)(c, a, d)(d, c, a) and so every product of disjoint transpositions is the product of 3 cycles. Lemma F.9.5 If n ≥ 5, then if B is a normal subgroup of A n , and B is not the identity, then B must contain a 3 cycle. Proof: Let α be the permutation in B which is “closest” to the identity without being the identity. That is, out of all permutations which are not the identity, this is one which has the most fixed points or equivalently moves the fewest numbers. Then α is the product of disjoint cycles. Suppose that the longest cycle is the first one and it has at least four numbers. Thus α =(i 1 ,i 2 ,i 3 ,i 4 , ··· ,m) γ 1 ···γ p
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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.4. FIRST ORDER LINEAR SYSTEMS 423
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Page 427 and 428: C.4. FIRST ORDER LINEAR SYSTEMS 427
Page 429 and 430: C.5. GEOMETRIC THEORY OF AUTONOMOUS
Page 431 and 432: C.5. GEOMETRIC THEORY OF AUTONOMOUS
Page 433 and 434: C.6. GENERAL GEOMETRIC THEORY 433 T
Page 435 and 436: C.7. THESTABLEMANIFOLD 435 Lemma C.
Page 437 and 438: C.7. THESTABLEMANIFOLD 437 ≤ e γ
Page 439 and 440: Compactness And Completeness D.0.1
Page 441 and 442: 441 |a k − a l | < ε 2 .Thenifk>
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
Page 449 and 450: F.2. THE FUNDAMENTAL THEOREM OF ALG
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: F.9. PERMUTATIONS 475 of f (x) ands
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 and 492: G.10. EXERCISES 491 11 It follows f
Page 493 and 494: G.13. EXERCISES 493 19 ⎛ ⎝ 4
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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