Linear Algebra

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$MATLAB C++ Math Library Reference$

272 Chapter 3. Maps Between Spaces 6 When the space shuttle Challenger exploded in 1986, one of the criticisms made of NASA’s decision to launch was in the way the analysis of number of O-ring failures versus temperature was made (of course, O-ring failure caused the explosion). Four O-ring failures will cause the rocket to explode. NASA had data from 24 previous flights. temp ◦ F 53 75 57 58 63 70 70 66 67 67 67 failures 3 2 1 1 1 1 1 0 0 0 0 68 69 70 70 72 73 75 76 76 78 79 80 81 0 0 0 0 0 0 0 0 0 0 0 0 0 The temperature that day was forecast to be 31 ◦ F. (a) NASA based the decision to launch partially on a chart showing only the flights that had at least one O-ring failure. Find the line that best fits these seven flights. On the basis of this data, predict the number of O-ring failures when the temperature is 31, and when the number of failures will exceed four. (b) Find the line that best fits all 24 flights. On the basis of this extra data, predict the number of O-ring failures when the temperature is 31, and when the number of failures will exceed four. Which do you think is the more accurate method of predicting? (An excellent discussion appears in [Dalal, et. al.].) 7 This table lists the average distance from the sun to each of the first seven planets, using earth’s average as a unit. Mercury Venus Earth Mars Jupiter Saturn Uranus 0.39 0.72 1.00 1.52 5.20 9.54 19.2 (a) Plot the number of the planet (Mercury is 1, etc.) versus the distance. Note that it does not look like a line, and so finding the line of best fit is not fruitful. (b) It does, however look like an exponential curve. Therefore, plot the number of the planet versus the logarithm of the distance. Does this look like a line? (c) The asteroid belt between Mars and Jupiter is thought to be what is left of a planet that broke apart. Renumber so that Jupiter is 6, Saturn is 7, and Uranus is 8, and plot against the log again. Does this look better? (d) Use least squares on that data to predict the location of Neptune. (e) Repeat to predict where Pluto is. (f) Is the formula accurate for Neptune and Pluto? This method was used to help discover Neptune (although the second item is misleading about the history; actually, the discovery of Neptune in position 9 prompted people to look for the “missing planet” in position 5). See [Gardner, 1970] 8 William Bennett has proposed an Index of Leading Cultural Indicators for the US ([Bennett], in 1993). Among the statistics cited are the average daily hours spent watching TV, and the average combined SAT scores. 1960 1965 1970 1975 1980 1985 1990 1992 TV 5:06 5:29 5:56 6:07 6:36 7:07 6:55 7:04 SAT 975 969 948 910 890 906 900 899 Suppose that a cause and effect relationship is proposed between the time spent watching TV and the decline in SAT scores (in this article, Mr. Bennett does not argue that there is a direct connection). (a) Find the line of best fit relating the independent variable of average daily TV hours to the dependent variable of SAT scores.
Topic: Line of Best Fit 273 (b) Find the most recent estimate of the average daily TV hours (Bennett’s cites Neilsen Media Research as the source of these estimates). Estimate the associated SAT score. How close is your estimate to the actual average? (Warning: a change has been made recently in the SAT, so you should investigate whether some adjustment needs to be made to the reported average to make a valid comparison.)
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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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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
Page 305 and 306: Section I. Definition 295 determine
Page 307 and 308: Section I. Definition 297 Exercises
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Page 311 and 312: Section I. Definition 301 That resu
Page 313 and 314: Section I. Definition 303 (b) Prove
Page 315 and 316: Section I. Definition 305 3.2 Defin
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Section II. Geometry of Determinant
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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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Topic: Projective Geometry 345 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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Section II. Similarity 363 3.19 Cor
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Section III. Nilpotence 365 5.III N
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Section III. Nilpotence 367 and thi
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Section III. Nilpotence 369 Proof.
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Section III. Nilpotence 371 The new
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Section III. Nilpotence 373 Now, wi
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Section III. Nilpotence 375 We can
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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 389 steps t
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Section IV. Jordan Form 391 each te
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Section IV. Jordan Form 393 2.13 Ex
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Section IV. Jordan Form 395 2.15 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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