Linear Algebra

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$MATLAB C++ Math Library Reference$

A-2 And. Consider the statement form ‘P and Q’. For the statement to be true both halves must hold: ‘7 is prime and so is 3’ is true, while ‘7 is prime and 3 is not’ is false. Here is the Venn diagram for ‘P and Q’. ✬✩ . . . . P Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✫✪ To prove ‘P and Q’, prove each half holds. Or. A‘P or Q’ is true when either half holds: ‘7 is prime or 4 is prime’ is true, while ‘7 is not prime or 4 is prime’ is false. We take ‘or’ inclusively so that if both halves are true ‘7 is prime or 4 is not’ then the statement as a whole is true. (In everyday speech sometimes ‘or’ is meant in an exclusive way: “Eat your vegetables or no dessert” does not intend both halves to hold.) The Venn diagram for ‘or’ includes all of both circles. ✬✩ . . . . P . . . . . . . Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ✫✪ . . . . . . . ✫✪ . . . . . . . . . . . . . . . . To prove ‘P or Q’ show that at all times at least one half holds (perhaps sometimes one half and sometimes the other, but at all times at least one). Implication. A‘P implies Q’ statement (perhaps instead phrased ‘if P then Q’ or‘P =⇒ Q’) is true unless P is true while Q is false. Thus ‘7 is prime implies 4 is not’ is true (contrary to its use in casual speech, in mathematics ‘if P then Q’ does not connote that P precedes Q or causes Q) while ‘7 is prime implies 4 is also prime’ is false. More subtly, ‘P =⇒ Q’ is always true when P is false: ‘4 is prime implies 7 is prime’ and ‘4 is prime implies 7 is not’ are both true statements, sometimes said to be vacuously true. (We adopt this convention because we want ‘if a number is a perfect square then it is not prime’ to be always true, for instance when the number is 5 or when the number is 6.) The diagram ✬✬✩✩ P Q ✫✫✪✪ shows Q holds whenever P does (another phrasing is ‘P is sufficient to give Q’). Notice again that if P does not hold, Q may or may not be in force. There are two main ways to establish an implication. The first way is direct: assume P is true and, using that assumption, prove Q. For instance, to show
A-3 ‘if a number is divisible by 5 then twice that number is divisible by 10’, assume the number is 5n and deduce that 2(5n) =10n. The second way is indirect: prove the contrapositive statement: ‘if Q is false then P is false’ (rephrased, ‘Q can only be false when P is also false’). As an example, to show ‘if a number is prime then it is not a perfect square’, argue that if it were a square p = n 2 then it could be factored p = n · n where 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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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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Chapter 4 Determinants In the first
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Section I. Definition 295 determine
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Section I. Definition 297 Exercises
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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 311 � 3.23
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Section I. Definition 317 (terms wi
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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 I. Complex Vector Spaces 34
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Section II. Similarity 351 5.II Sim
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Section II. Similarity 353 1.7 Cons
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Section II. Similarity 355 2.4 Coro
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Section II. Similarity 357 2.12 The
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Section II. Similarity 359 and the
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Section II. Similarity 361 Proof. A
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Page 409 and 410: Topic: Computing Eigenvalues—the
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Page 413 and 414: Topic: Stable Populations 403 Topic
Page 415 and 416: Topic: Linear Recurrences 405 Topic
Page 417 and 418: Topic: Linear Recurrences 407 And,
Page 419 and 420: Topic: Linear Recurrences 409 diamo
Page 421 and 422: Topic: Linear Recurrences 411 Compu
Page 423: Appendix Introduction Mathematics i
Page 427 and 428: Techniques of Proof A-5 Induction.
Page 429 and 430: A-7 The Principle of Extensionality
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Page 435 and 436: Bibliography [Abbot] Edwin Abbott,
Page 437 and 438: [Feller] William Feller, An Introdu
Page 439 and 440: [Programmer’s Ref.] Microsoft Pro
Page 441 and 442: congruent plane figures, 286 contra
Page 443 and 444: Gauss’ method, 3 Gauss-Jordan, 46
Page 445 and 446: educed echelon form, 46 reflection,
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Linear Algebra

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