Linear Algebra

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$MATLAB C++ Math Library Reference$

324 Chapter 4. Determinants � 1.19 Let T be the matrix representing (with respect to the standard bases) the map that rotates plane vectors counterclockwise thru θ radians. By what factor does T change sizes? � 1.20 Must a transformation t: R 2 → R 2 that preserves areas also preserve lengths? � 1.21 What is the volume of a parallelepiped in R 3 bounded by a linearly dependent set? � 1.22 Find the area of the triangle in R 3 with endpoints (1, 2, 1), (3, −1, 4), and (2, 2, 2). (Area, not volume. The triangle defines a plane—what is the area of the triangle in that plane?) � 1.23 An alternate proof of Theorem 1.5 uses the definition of determinant functions. (a) Note that the vectors forming S make a linearly dependent set if and only if |S| = 0, and check that the result holds in this case. (b) For the |S| �= 0 case, to show that |TS|/|S| = |T | for all transformations, consider the function d: Mn×n → R given by T ↦→ |TS|/|S|. Show that d has the first property of a determinant. (c) Show that d has the remaining three properties of a determinant function. (d) Conclude that |TS| = |T |·|S|. 1.24 Give a non-identity matrix with the property that A trans = A −1 . Show that if A trans = A −1 then |A| = ±1. Does the converse hold? 1.25 The algebraic property of determinants that factoring a scalar out of a single row will multiply the determinant by that scalar shows that where H is 3×3, the determinant of cH is c 3 times the determinant of H. Explain this geometrically, that is, using Theorem 1.5, � 1.26 Matrices H and G are said to be similar if there is a nonsingular matrix P such that H = P −1 GP (we will study this relation in Chapter Five). Show that similar matrices have the same determinant. 1.27 We usually represent vectors in R 2 with respect to the standard basis so vectors in the first quadrant have both coordinates positive. � � �v +3 RepE2 (�v) = +2 Moving counterclockwise around the origin, we cycle thru four regions: � � � � � � � � + − − + ··· −→ −→ −→ −→ −→ ··· . + + − − Using this basis � � 0 B = 〈 , 1 � � −1 〉 0 β2 � gives the same counterclockwise cycle. We say these two bases have the same orientation. (a) Whydotheygivethesamecycle? (b) What other configurations of unit vectors on the axes give the same cycle? (c) Find the determinants of the matrices formed from those (ordered) bases. (d) What other counterclockwise cycles are possible, and what are the associated determinants? (e) What happens in R 1 ? (f) What happens in R 3 ? A fascinating general-audience discussion of orientations is in [Gardner]. �β 1
Section II. Geometry of Determinants 325 1.28 This question uses material from the optional Determinant Functions Exist subsection. Prove Theorem 1.5 by using the permutation expansion formula for the determinant. � 1.29 (a) Show that this gives the equation of a line in R 2 thru (x2,y2)and(x3,y3). � � �x x2 x3� � � �y y2 y3� � 1 1 1 � =0 (b) [Petersen] Prove that the area of a triangle with vertices (x1,y1), (x2,y2), and (x3,y3) is � �x1 1 � �y1 2 � 1 x2 y2 1 � x3� � y3� 1 � . (c) [Math. Mag., Jan. 1973] Prove that the area of a triangle with vertices at (x1,y1), (x2,y2), and (x3,y3) whose coordinates are integers has an area of N or N/2 for some positive integer N.
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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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6 Chapter 1. Linear Systems the sec
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8 Chapter 1. Linear Systems Don’t
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10 Chapter 1. Linear Systems 1.25 G
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12 Chapter 1. Linear Systems can al
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14 Chapter 1. Linear Systems Matric
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16 Chapter 1. Linear Systems Scalar
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18 Chapter 1. Linear Systems Must t
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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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26 Chapter 1. Linear Systems shows
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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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44 Chapter 1. Linear Systems 2.25 P
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46 Chapter 1. Linear Systems We can
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48 Chapter 1. Linear Systems each e
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54 Chapter 1. Linear Systems x1 has
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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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64 Chapter 1. Linear Systems For th
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66 Chapter 1. Linear Systems (a) Fi
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68 Chapter 1. Linear Systems 4 3 2
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72 Chapter 1. Linear Systems Topic:
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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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120 Chapter 2. Vector Spaces we stu
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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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142 Chapter 2. Vector Spaces We cou
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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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Page 303 and 304: Chapter 4 Determinants In the first
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Section III. Nilpotence 377 � −
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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 383 Equate
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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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educed echelon form, 46 reflection,
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Linear Algebra

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