Linear Algebra

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$MATLAB C++ Math Library Reference$

262 Chapter 3. Maps Between Spaces (Verification that R 3 = M ⊕ N and R 3 = M ⊕ ˆ N is routine.) We will check that these projections are different by checking that they have different effects on this vector. ⎛ �v = ⎝ 2 ⎞ 2⎠ 5 For the first one we find a basis for N ⎛ BN = 〈 ⎝ 0 ⎞ 0⎠〉 1 ⌢ and represent �v with respect to the concatenation BM BN . ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 1 0 0 ⎝2⎠ =2· ⎝0⎠ +1· ⎝2⎠ +4· ⎝0⎠ 5 0 1 1 The projection of �v into M along N is found by keeping the M part and dropping the N part. ⎛ ⎞ 1 ⎛ ⎞ 0 ⎛ ⎞ 2 projM,N(�v )=2· ⎝0⎠ +1· ⎝2⎠ = ⎝2⎠ 0 1 1 For the other subspace ˆ N, this basis is natural. ⎛ ⎞ 0 BN ˆ = 〈 ⎝ 1 ⎠〉 −2 Representing �v with respect to the concatenation ⎛ ⎞ 2 ⎛ ⎞ 1 ⎛ ⎞ 0 ⎛ ⎞ 0 ⎝2⎠ =2· ⎝0⎠ +(9/5) · ⎝2⎠ − (8/5) · ⎝ 1 ⎠ 5 0 1 −2 and then keeping only the M part gives this. ⎛ projM, N ˆ (�v )=2· ⎝ 1 ⎞ ⎛ 0⎠ +(9/5) · ⎝ 0 0 ⎞ ⎛ 2⎠ = ⎝ 1 2 ⎞ 18/5⎠ 9/5 Therefore projection along different subspaces may yield different results. These pictures compare the two maps. Both show that the projection is indeed ‘into’ the plane and ‘along’ the line.
Section VI. Projection 263 N M Notice that the projection along N is not orthogonal—there are members of the plane M that are not orthogonal to the dotted line. But the projection along ˆN is orthogonal. A natural question is: what is the relationship between the projection operation defined above, and the operation of orthogonal projection into a line? The second picture above suggests the answer—orthogonal projection into a line is a special case of the projection defined above; it is just projection along a subspace perpendicular to the line. N In addition to pointing out that projection along a subspace is a generalization, this scheme shows how to define orthogonal projection into any subspace of R n , of any dimension. 3.4 Definition The orthogonal complement of a subspace M of R n is M ⊥ = {�v ∈ R n � � �v is perpendicular to all vectors in M} (read “M perp”). The orthogonal projection proj M (�v ) of a vector is its projection into M along M ⊥ . 3.5 Example In R3 , to find the orthogonal complement of the plane ⎛ P = { ⎝ x ⎞ y⎠ z � � 3x +2y − z =0} we start with a basis for P . ˆN M ⎛ B = 〈 ⎝ 1 ⎞ ⎛ 0⎠ , ⎝ 3 0 ⎞ 1⎠〉 2 Any �v perpendicular to every vector in B is perpendicular to every vector in the span of B (the proof of this assertion is Exercise 19). Therefore, the subspace M
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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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70 Chapter 1. Linear Systems expert
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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 307 and 308: Section I. Definition 297 Exercises
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Section I. Definition 313 could be
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Section I. Definition 315 We still
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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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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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