Linear Algebra

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$MATLAB C++ Math Library Reference$

248 Chapter 3. Maps Between Spaces � 1 (c) 2 � � 3 1 , 6 2 � 3 −6 � 2.11 Find the canonical representative of the matrix-equivalence class of each matrix. (a) � 2 4 1 2 � 0 0 (b) � 0 1 3 1 1 3 0 0 3 � 2 4 −1 2.12 Suppose that, with respect to B = E2 � � � � 1 1 D = 〈 , 〉 1 −1 the transformation t: R 2 → R 2 is represented by this matrix. � � 1 2 3 4 Use change of basis matrices to represent t with respect to each pair. (a) ˆ � � � � 0 1 B = 〈 , 〉, 1 1 ˆ � � � � −1 2 D = 〈 , 〉 0 1 (b) ˆ � � � � 1 1 B = 〈 , 〉, 2 0 ˆ � � � � 1 2 D = 〈 , 〉 2 1 � 2.13 What size are P and Q? � 2.14 Use Theorem 2.6 to show that a square matrix is nonsingular if and only if it is equivalent to an identity matrix. � 2.15 Show that, where A is a nonsingular square matrix, if P and Q are nonsingular square matrices such that PAQ = I then QP = A −1 . � 2.16 Why does Theorem 2.6 not show that every matrix is diagonalizable (see Example 2.2)? 2.17 Must matrix equivalent matrices have matrix equivalent transposes? 2.18 What happens in Theorem 2.6 if k =0? � 2.19 Show that matrix-equivalence is an equivalence relation. � 2.20 Show that a zero matrix is alone in its matrix equivalence class. Are there other matrices like that? 2.21 What are the matrix equivalence classes of matrices of transformations on R 1 ? R 3 ? 2.22 How many matrix equivalence classes are there? 2.23 Are matrix equivalence classes closed under scalar multiplication? Addition? 2.24 Let t: R n → R n represented by T with respect to En, En. (a) Find RepB,B(t) in this specific case. � � � � � � 1 1 1 −1 T = B = 〈 , 〉 3 −1 2 −1 (b) Describe Rep B,B(t) in the general case where B = 〈 � β1,... , � βn〉. 2.25 (a) Let V have bases B1 and B2 and suppose that W has the basis D. Where h: V → W , find the formula that computes Rep B2,D(h) fromRep B1,D(h). (b) Repeat the prior question with one basis for V and two bases for W . 2.26 (a) If two matrices are matrix-equivalent and invertible, must their inverses be matrix-equivalent? (b) If two matrices have matrix-equivalent inverses, must the two be matrixequivalent?
Section V. Change of Basis 249 (c) If two matrices are square and matrix-equivalent, must their squares be matrix-equivalent? (d) If two matrices are square and have matrix-equivalent squares, must they be matrix-equivalent? � 2.27 Square matrices are similar if they represent the same transformation, but each with respect to the same ending as starting basis. That is, RepB1,B1 (t) is similar to Rep B2,B2 (t). (a) Give a definition of matrix similarity like that of Definition 2.3. (b) Prove that similar matrices are matrix equivalent. (c) Show that similarity is an equivalence relation. (d) Show that if T is similar to ˆ T then T 2 is similar to ˆ T 2 , the cubes are similar, etc. Contrast with the prior exercise. (e) Prove that there are matrix equivalent matrices that are not similar.
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Preface In most mathematics program
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dent projects for individuals or sm
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Contents 1 Linear Systems 1 1.I Sol
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5.II.1 Definition and Examples ....
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2 Chapter 1. Linear Systems To fini
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4 Chapter 1. Linear Systems Conside
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14 Chapter 1. Linear Systems Matric
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16 Chapter 1. Linear Systems Scalar
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20 Chapter 1. Linear Systems 2.30 [
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22 Chapter 1. Linear Systems Studyi
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24 Chapter 1. Linear Systems the bo
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28 Chapter 1. Linear Systems 3.13 E
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30 Chapter 1. Linear Systems (a) 2x
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32 Chapter 1. Linear Systems 1.II L
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34 Chapter 1. Linear Systems positi
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36 Chapter 1. Linear Systems In R3
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38 Chapter 1. Linear Systems � 1.
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40 Chapter 1. Linear Systems Notice
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42 Chapter 1. Linear Systems 2.7 De
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48 Chapter 1. Linear Systems each e
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56 Chapter 1. Linear Systems First,
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58 Chapter 1. Linear Systems 2.8 Ex
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60 Chapter 1. Linear Systems (2) Ca
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62 Chapter 1. Linear Systems (b) Th
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74 Chapter 1. Linear Systems We beg
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76 Chapter 1. Linear Systems 1 Calc
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78 Chapter 1. Linear Systems parall
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80 Chapter 2. Vector Spaces 2.I Def
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82 Chapter 2. Vector Spaces For the
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84 Chapter 2. Vector Spaces A vecto
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86 Chapter 2. Vector Spaces 1.12 Ex
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88 Chapter 2. Vector Spaces Chapter
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90 Chapter 2. Vector Spaces (d) The
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92 Chapter 2. Vector Spaces 2.2 Exa
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94 Chapter 2. Vector Spaces second.
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96 Chapter 2. Vector Spaces Proof.
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98 Chapter 2. Vector Spaces The sub
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100 Chapter 2. Vector Spaces S. Mem
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102 Chapter 2. Vector Spaces 2.II L
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104 Chapter 2. Vector Spaces 1.5 Ex
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106 Chapter 2. Vector Spaces Lookin
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108 Chapter 2. Vector Spaces Proof.
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110 Chapter 2. Vector Spaces (a) {2
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112 Chapter 2. Vector Spaces � 1.
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114 Chapter 2. Vector Spaces 1.5 De
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116 Chapter 2. Vector Spaces 1.13 D
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118 Chapter 2. Vector Spaces � 1.
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122 Chapter 2. Vector Spaces 2.11 C
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124 Chapter 2. Vector Spaces subspa
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126 Chapter 2. Vector Spaces 3.6 De
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128 Chapter 2. Vector Spaces Anothe
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130 Chapter 2. Vector Spaces (a)
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132 Chapter 2. Vector Spaces by fin
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134 Chapter 2. Vector Spaces has a
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136 Chapter 2. Vector Spaces 4.12 E
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138 Chapter 2. Vector Spaces 4.22 I
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140 Chapter 2. Vector Spaces (c) Le
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144 Chapter 2. Vector Spaces Then w
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146 Chapter 2. Vector Spaces (d) Pl
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148 Chapter 2. Vector Spaces howeve
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150 Chapter 2. Vector Spaces positi
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152 Chapter 2. Vector Spaces Topic:
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154 Chapter 2. Vector Spaces The cl
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156 Chapter 2. Vector Spaces Remark
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158 Chapter 2. Vector Spaces (b) Sh
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160 Chapter 3. Maps Between Spaces
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Section I. Definition 299 1.15 Prov
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Section I. Definition 301 That resu
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Section I. Definition 303 (b) Prove
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Section I. Definition 305 3.2 Defin
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Section I. Definition 307 Therefore
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Section I. Definition 315 We still
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Section II. Geometry of Determinant
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Section III. Other Formulas 327 The
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Section III. Other Formulas 329 1.9
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Topic: Cramer’s Rule 331 Topic: C
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Topic: Cramer’s Rule 333 4 Suppos
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Topic: Speed of Calculating Determi
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Topic: Projective Geometry 337 Topi
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Topic: Projective Geometry 339 P P
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Topic: Projective Geometry 341 of t
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Topic: Projective Geometry 343 The
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Chapter 5 Similarity While studying
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Section II. Similarity 351 5.II Sim
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Section II. Similarity 361 Proof. A
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Section III. Nilpotence 369 Proof.
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Section IV. Jordan Form 379 5.IV Jo
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Section IV. Jordan Form 381 Using t
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Section IV. Jordan Form 385 � 1.1
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Section IV. Jordan Form 387 is (x
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Section IV. Jordan Form 393 2.13 Ex
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Section IV. Jordan Form 397 � 5 4
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Topic: Computing Eigenvalues—the
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Topic: Computing Eigenvalues—the
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Topic: Stable Populations 403 Topic
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Topic: Linear Recurrences 405 Topic
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Topic: Linear Recurrences 407 And,
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Topic: Linear Recurrences 409 diamo
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Topic: Linear Recurrences 411 Compu
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Appendix Introduction Mathematics i
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A-3 ‘if a number is divisible by
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Techniques of Proof A-5 Induction.
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A-7 The Principle of Extensionality
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A-9 So an identity map plays the sa
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A-11 Similarly, the equivalence rel
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Bibliography [Abbot] Edwin Abbott,
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[Feller] William Feller, An Introdu
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[Programmer’s Ref.] Microsoft Pro
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congruent plane figures, 286 contra
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Gauss’ method, 3 Gauss-Jordan, 46
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Linear Algebra

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