Linear Algebra, Theory And Applications, 2012a

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72 MATRICES AND LINEAR TRANSFORMATIONS 5. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of 2π/3. 6. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of π/12. Hint: Note that π/12 = π/3 − π/4. 7. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of 2π/3 and then reflects across the x axis. 8. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of π/3 and then reflects across the x axis. 9. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of π/4 and then reflects across the x axis. 10. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of π/6 and then reflects across the x axis followed by a reflection across the y axis. 11. Find the matrix for the linear transformation which reflects every vector in R 2 across the x axis and then rotates every vector through an angle of π/4. 12. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of π/4 and next reflects every vector across the x axis. Compare with the above problem. 13. Find the matrix for the linear transformation which reflects every vector in R 2 across the x axis and then rotates every vector through an angle of π/6. 14. Find the matrix for the linear transformation which reflects every vector in R 2 across the y axis and then rotates every vector through an angle of π/6. 15. Find the matrix for the linear transformation which rotates every vector in R 2 through an angle of 5π/12. Hint: Note that 5π/12 = 2π/3 − π/4. 16. Find the matrix for proj u (v) whereu =(1, −2, 3) T . 17. Find the matrix for proj u (v) whereu =(1, 5, 3) T . 18. Find the matrix for proj u (v) whereu =(1, 0, 3) T . 19. Give an example of a 2 × 2 matrix A which has all its entries nonzero and satisfies A 2 = A. A matrix which satisfies A 2 = A is called idempotent. 20. Let A be an m × n matrix and let B be an n × m matrix where n
2.7. EXERCISES 73 22. If A is a linear transformation, and Ax p = b, show that the general solution to the equation Ax = b is of the form x p + y where y ∈ ker (A). By this I mean to show that whenever Az = b there exists y ∈ ker (A) such that x p + y = z. For the definition of ker (A) see Problem 21. 23. Using Problem 21, find the general solution to the following linear system. ⎛ ⎜ ⎝ 1 2 3 2 1 0 2 1 1 2 1 4 4 3 3 0 2 1 1 2 ⎛ ⎞ ⎟ ⎠ ⎜ ⎝ x 1 x 2 x 3 x 4 x 5 ⎞ ⎛ ⎟ ⎠ = ⎜ ⎝ 24. Using Problem 21, find the general solution to the following linear system. ⎛ ⎜ ⎝ 1 2 3 2 1 0 2 1 1 2 1 4 4 3 3 0 2 1 1 2 ⎛ ⎞ ⎟ ⎠ ⎜ ⎝ x 1 x 2 x 3 x 4 x 5 ⎞ ⎛ ⎟ ⎠ = ⎜ ⎝ 25. Show that the function T u defined by T u (v) ≡ v − proj u (v) is also a linear transformation. 26. If u =(1, 2, 3) T , as in Example 9.3.22 and T u is given in the above problem, find the matrix A u which satisfies A u x = T u (x). 27. Suppose V is a subspace of F n and T : V → F p is a nonzero linear transformation. Show that there exists a basis for Im (T ) ≡ T (V ) {T v 1 , ··· ,Tv m } 11 7 18 7 6 7 13 7 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ and that in this situation, is linearly independent. {v 1 , ··· , v m } 28. ↑In the situation of Problem 27 where V is a subspace of F n , show that there exists {z 1 , ··· , z r } a basis for ker (T ) . (Recall Theorem 2.4.12. Since ker (T ) is a subspace, it has a basis.) Now for an arbitrary T v ∈ T (V ) , explain why and why this implies T v = a 1 T v 1 + ···+ a m T v m v − (a 1 v 1 + ···+ a m v m ) ∈ ker (T ) . Then explain why V =span(v 1 , ··· , v m , z 1 , ··· , z r ) . 29. ↑In the situation of the above problem, show {v 1 , ··· , v m , z 1 , ··· , z r } is a basis for V and therefore, dim (V )=dim(ker(T )) + dim (T (V )) .
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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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Page 23 and 24: 1.9. DIVISION AND NUMBERS 23 Theore
Page 25 and 26: 1.9. DIVISION AND NUMBERS 25 Exampl
Page 27 and 28: 1.10. SYSTEMS OF EQUATIONS 27 Why i
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
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Page 71: 2.7. EXERCISES 71 this will suffice
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Page 77 and 78: Determinants 3.1 Basic Techniques A
Page 79 and 80: 3.1. BASIC TECHNIQUES AND PROPERTIE
Page 81 and 82: 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
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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
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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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