Linear Algebra

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$MATLAB C++ Math Library Reference$

160 Chapter 3. Maps Between Spaces then this correspondence preserves the operations, for instance this addition � 1 2 � + � 3 4 � = � 4 6 � ←→ and this scalar multiplication. 5 · � 1 2 � = � 5 10 � ←→ 5 · More generally stated, under the correspondence both operations are preserved: and � � a0 a1 ←→ � � 1 + 2 � � a0 � � � � � � a0 a1 + b0 b1 = a0 + b0 a1 + b1 ←→ a1 r · � � � � a0 a1 = ra0 ra1 ←→ r · (all of the variables are real numbers). � � 1 = 2 � � 3 = 4 � � 5 10 � � 4 6 � � � � a0 b0 + = a1 b1 � � � � a0 ra0 = a1 ra1 � a0 + b0 a1 + b1 1.2 Example Another two spaces we can think of as “the same” are P2, the space of quadratic polynomials, and R 3 . A natural correspondence is this. a0 + a1x + a2x 2 ←→ ⎛ ⎝ a0 a1 a2 ⎞ ⎠ (e.g., 1 + 2x +3x2 ⎛ ⎞ 1 ←→ ⎝2⎠) 3 The structure is preserved: corresponding elements add in a corresponding way a0 + a1x + a2x 2 + b0 + b1x + b2x 2 (a0 + b0)+(a1 + b1)x +(a2 + b2)x 2 and scalar multiplication corresponds also. ←→ ⎛ ⎝ a0 ⎞ ⎛ ⎠ + ⎝ b0 ⎞ ⎛ ⎠ = r · (a0 + a1x + a2x 2 )=(ra0)+(ra1)x +(ra2)x 2 ←→ r · a1 a2 b1 b2 � ⎝ a0 + b0 a1 + b1 a2 + b2 ⎞ ⎠ ⎛ ⎝ a0 ⎞ ⎛ a1⎠ = ⎝ a2 ra0 ⎞ ra1⎠ ra2
Section I. Isomorphisms 161 1.3 Definition An isomorphism between two vector spaces V and W is a map f : V → W that (1) is a correspondence: f is one-to-one and onto; ∗ (2) preserves structure: if �v1,�v2 ∈ V then and if �v ∈ V and r ∈ R then f(�v1 + �v2) =f(�v1)+f(�v2) f(r�v) =rf(�v) (we write V ∼ = W ,read“V is isomorphic to W ”, when such a map exists). (“Morphism” means map, so “isomorphism” means a map expressing sameness.) 1.4 Example The vector space G = {c1 cos θ + c2 sin θ � � c1,c2 ∈ R} of functions of θ is isomorphic to the vector space R2 under this map. c1 cos θ + c2 sin θ f � � c1 ↦−→ We will check this by going through the conditions in the definition. We will first verify condition (1), that the map is a correspondence between the sets underlying the spaces. To establish that f is one-to-one, we must prove that f(�a) =f( � b) only when �a = � b.If then, by the definition of f, f(a1 cos θ + a2 sin θ) =f(b1 cos θ + b2 sin θ) � � � � a1 b1 = a2 from which we can conclude that a1 = b1 and a2 = b2 because column vectors are equal only when they have equal components. We’ve proved that f(�a) =f( �b) implies that �a = �b, which shows that f is one-to-one. To check that f is onto we must check that any member of the codomain R2 mapped to. But that’s clear—any � � x ∈ R y 2 is the image, under f, of this member of the domain: x cos θ + y sin θ ∈ G. Next we will verify condition (2), that f preserves structure. ∗ More information on one-to-one and onto maps is in the appendix. b2 c2
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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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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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42 Chapter 1. Linear Systems 2.7 De
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56 Chapter 1. Linear Systems First,
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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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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 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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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 IV. Jordan Form 379 5.IV Jo
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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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