Linear Algebra, Theory And Applications, 2012a

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490 ANSWERS TO SELECTED EXERCISES 6 ⎛ ⎝ ⎞ 1 2 e−t 1 0 2 e−t 1 2 cos t + 1 2 sin t − sin t 1 2 sin t − 1 2 cos t ⎠ 1 2 sin t − 1 2 cos t cos t − 1 2 cos t − 1 2 sin t 8 det(λI − A) =det ( λI − S −1 BS ) =det ( λS −1 S − S −1 BS ) =det ( S −1 (λI − B) S ) =det ( S −1) det (λI − B)det(S) =det ( S −1 S ) det (λI − B) =det(λI − B) 9 From the Cayley Hamilton theorem,A n +a n−1 A n−1 + ···+ a 1 A + a 0 I =0. Also the characteristic polynomial is det (tI − A) and the constant term is (−1) n det (A) . Thus a 0 ≠0ifandonlyifdet(A) ≠ 0 if and only if A −1 has an inverse. Thus if A −1 exists, it follows that a 0 I = − ( A n + a n−1 A n−1 + ···+ a 1 A ) = A ( −A n−1 − a n−1 A n−2 −···−a 1 I ) and also a 0 I = ( −A n−1 − a n−1 A n−2 −···−a 1 I ) A Therefore, the inverse is ( 1 a 0 −A n−1 − a n−1 A n−2 −···−a 1 I ) 11 Say the characteristic polynomial is q (t) whichisof degree 3. Then if n ≥ 3,t n = q (t) l (t)+r (t) where the degree of r (t) is either less than 3 or it equals zero. Thus A n = q (A) l (A) +r (A) =r (A) and so all the terms A n for n ≥ 3 can be replaced with some r (A) where the degree of r (t) is no more than 2. Thus, assuming there are no convergence issues, the infinite sum must be of the form ∑ 2 k=0 b kA k . G.9 Exercises 4.6 1 A typical thing in {Ax : x ∈ P (u 1 , ··· , u n )} is ∑ n k=1 t kAu k : t k ∈ [0, 1] and so it is just P (Au 1 , ··· ,Au n ) . ( ) 1 1 2 E = 0 1 P (e 1 , e 2 ) E(P (e 1 , e 2 )) 5Heretheyare. ⎛ 1 0 0 ⎞ ⎛ ⎝ 0 1 0 ⎠ , ⎝ ⎛ 0 0 1 0 1 0 ⎞ ⎛ ⎝ 1 0 0 ⎠ , ⎝ 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 0 ⎞ ⎛ ⎠ , ⎝ ⎞ ⎛ ⎠ , ⎝ 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 ⎞ ⎠ ⎞ ⎠ So what is the dimension of the span of these? One way to systematically accomplish this is to unravel them and then use the row reduced echelon form. Unraveling these yields the column vectors ⎛ ⎜ ⎝ 1 0 0 0 1 0 0 0 1 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 0 0 1 1 0 0 0 1 0 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 0 1 0 0 0 1 1 0 0 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 0 1 0 1 0 0 0 0 1 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 1 0 0 0 0 1 0 1 0 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 0 0 1 0 1 0 1 0 0 ⎞ ⎟ ⎠ Then arranging these as the columns of a matrix yields the following along with its row reduced echelon form. ⎛ ⎞ 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 1 ,rowechelonform: 0 0 1 0 1 0 ⎜ 0 0 1 0 0 1 ⎟ ⎝ 0 1 0 0 1 0 ⎠ 1 0 0 1 0 0 ⎛ ⎜ ⎝ 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 −1 0 0 0 0 1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The dimension is 5. ⎞ ⎟ ⎠ 10 It is because you cannot have more than min (m, n) nonzero rows in the row reduced echelon form. Recall that the number of pivot columns is the same as the number of nonzero rows from the description of this row reduced echelon form.
G.10. EXERCISES 491 11 It follows from the fact that e 1 , ··· , e m occur as columns in row reduced echelon form that the dimension of the column space of A is n and so, since this column space is A (R n ) , it follows that it equals F m . 12 Since m>nthe dimension of the column space of A is no more than n and so the columns of A cannot span F m . 15 If ∑ ∑ i c iz i = 0, apply A to both sides to obtain i c iw i = 0. By assumption, each c i =0. 19 There are more columns than rows and at most m can be pivot columns so it follows at least one column is a linear combination of the others hence A is not one too one. 21 |b−Ay| 2 = |b−Ax+Ax−Ay| 2 = |b−Ax| 2 + |Ax − Ay| 2 +2(b−Ax,A (x − y)) = |b−Ax| 2 +|Ax − Ay| 2 +2 ( A T b−A T Ax, (x − y) ) = |b−Ax| 2 + |Ax − Ay| 2 and so, Ax is closest to b out of all vectors Ay. 27 No. ⎛ ⎜ ⎝ 1 0 2 0 0 1 1 7 0 0 0 1 0 0 0 0 ⎞ ⎟ ⎠ 29 Let A be an m × n matrix. Then ker (A) is a subspace of F n . Is it true that every subspace of F n is the kernel or null space of some matrix? Prove or disprove. Let M be a subspace of F n . If it equals {0} , consider the matrix I. Otherwise, it has a basis {m 1 , ··· , m k } . Consider the matrix ( m1 ··· m k 0 ) where 0 is either not there in case k = n or has n − k columns. 30 This is easy to see when you consider that P ij is its own inverse and that P ij multiplied on the right switches the i th and j th columns. Thus you switch the columns and then you switch the rows. This has the effect of switching A ii and A jj . For example, ⎛ ⎜ ⎝ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ a b c d e f z h j k l m n t h g ⎞ ⎟ ⎠ · ⎛ ⎜ ⎝ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ⎞ ⎛ ⎟ ⎠ = ⎜ ⎝ a d c b n g h t j m l k e h z f ⎞ ⎟ ⎠ More formally, the ii th entry of P ij AP ij is ∑ P ij is A spP ij pi = P ij ij A jjP ij ji = A ij s,p 31 If A has an inverse, then it is one to one. Hence the columns are independent. Therefore, they are each pivot columns. Therefore, the row reduced echelon form of A is I. This is what was needed for the procedure to work. G.10 Exercises 5.8 1 ⎛ ⎝ 1 2 2 1 0 3 ⎞ ⎠ . = ⎛ 1 2 3 ⎞ ⎛ ⎞ 1 0 0 1 2 0 ⎝ 2 1 0 ⎠ ⎝ 0 −3 3 ⎠ 1 0 1 0 0 3 ⎛ ⎞ ⎛ ⎞ 3 ⎝ 1 2 1 1 2 2 ⎠ . = ⎝ 1 0 0 0 0 1 ⎠ · ⎛ 2 1 1 ⎞ ⎛ 0 1 0 ⎞ ⎝ 1 0 0 2 1 0 ⎠ ⎝ 1 2 1 0 −3 −1 ⎠ 1 0 1 0 0 1 ⎛ ⎞ ⎛ ⎞ 1 2 1 1 0 0 0 5 ⎜ 1 2 2 ⎟ ⎝ 2 4 1 ⎠ . = ⎜ 0 0 0 1 ⎟ ⎝ 0 0 1 0 ⎠ · 3 2 1 0 1 0 0 ⎛ ⎞ ⎛ ⎞ 1 0 0 0 1 2 1 ⎜ 3 1 0 0 ⎟ ⎜ 0 −4 −2 ⎟ ⎝ 2 0 1 0 ⎠ ⎝ 0 0 −1 ⎠ 1 0 −1 1 0 0 0 ⎛ ⎞ 9 ⎝ 1 2 1 0 3 0 1 1 ⎠ 1 0 2 1 ⎛ √ √ ⎞ 1 11 11 1 √ 11√ √ 10 √ 11 0 √ = ⎝ 3 11 11 − 3 110 10 11 − 1 10√ 2 5 ⎠ √ √ √ √ √ · 1 11 11 − 1 110 10 11 3 10 2 5 ⎛ √ √ √ 11 2 11 11 6 11 11 4 √ √ √ √ 11√ 11 √ ⎝ 2 0 11 10 11 1 22 10 11 − 2 √ √ 55√ √ 10 √ 11 1 0 0 2 2 5 1 5 2 5 ⎞ ⎠
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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
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 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: G.7. EXERCISES 489 15 Find the matr
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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Linear Algebra, Theory And Applications, 2012a

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