Linear Algebra

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$MATLAB C++ Math Library Reference$

196 Chapter 3. Maps Between Spaces 1.2 Definition Suppose that V and W are vector spaces of dimensions n and m with bases B and D, and that h: V → W is a linear map. If RepD(h( � ⎛ ⎞ h1,1 ⎜ h2,1 ⎟ ⎜ ⎟ β1)) = ⎜ . ⎟ , ... ,Rep ⎝ . ⎠ D(h( � ⎛ ⎞ h1,n ⎜ h2,n ⎟ ⎜ ⎟ βn)) = ⎜ . ⎟ ⎝ . ⎠ then hm,1 ⎛ ⎜ RepB,D(h) = ⎜ ⎝ D h1,1 h1,2 ... h1,n h2,1 h2,2 ... h2,n . hm,1 hm,2 ... hm,n is the matrix representation of h with respect to B,D. ⎞ ⎟ ⎠ hm,n Briefly, the vectors representing the h( � β)’s are adjoined to make the matrix representing the map. ⎛ . ⎜ . . RepB,D(h) = ⎜ ⎝ RepD( h( � β1)) ··· RepD( h( � ⎞ ⎟ βn)) ⎟ ⎠ . . . . Observe that the number of columns of the matrix is the dimension of the domain of the map, and the number of rows is the dimension of the codomain. 1.3 Example If h: R3 →P1 is given by ⎛ ⎞ a1 ⎝ ⎠ h ↦−→ (2a1 + a2)+(−a3)x then where a2 a3 B,D ⎛ ⎞ 0 ⎛ ⎞ 0 ⎛ ⎞ 2 B = 〈 ⎝0⎠ , ⎝2⎠ , ⎝0⎠〉 and D = 〈1+x, −1+x〉 1 0 0 the action of h on B is given by ⎛ ⎝ 0 ⎞ 0⎠ 1 h ↦−→ −x and a simple calculation gives � � −1/2 RepD(−x) = −1/2 D ⎛ ⎝ 0 ⎞ 2⎠ 0 h ↦−→ 2 Rep D(2) = � � 1 −1 D ⎛ ⎝ 2 ⎞ 0⎠ 0 h ↦−→ 4 Rep D(4) = D � 2 −2 � D
Section III. Computing <strong>Linear</strong> Maps 197 showing that this is the matrix representing h with respect to the bases. Rep B,D(h) = � −1/2 1 � 2 −1/2 −1 −2 B,D We will use lower case letters for a map, upper case for the matrix, and lower case again for the entries of the matrix. Thus for the map h, the matrix representing it is H, with entries hi,j. 1.4 Theorem Assume that V and W are vector spaces of dimensions m and n with bases B and D, and that h: V → W is a linear map. If h is represented by ⎛ ⎞ h1,1 h1,2 ... h1,n ⎜ h2,1 h2,2 ⎜ ... h2,n ⎟ RepB,D(h) = ⎜ . ⎝ . ⎟ . ⎠ hm,1 hm,2 ... hm,n and �v ∈ V is represented by ⎛ ⎞ c1 ⎜c2⎟ ⎜ ⎟ RepB(�v) = ⎜ ⎝ . ⎟ . ⎠ cn B B,D then the representation of the image of �v is this. ⎛ ⎞ h1,1c1 + h1,2c2 + ···+ h1,ncn ⎜ h2,1c1 ⎜ + h2,2c2 + ···+ h2,ncn ⎟ RepD( h(�v))= ⎜ ⎝ . ⎟ . ⎠ hm,1c1 + hm,2c2 + ···+ hm,ncn Proof. Exercise 28. QED We will think of the matrix Rep B,D(h) and the vector Rep B(�v) as combining to make the vector Rep D(h(�v)). 1.5 Definition The matrix-vector product of a m×n matrix and a n×1 vector is this. ⎛ ⎞ ⎛ ⎞ a1,1 a1,2 ... a1,n ⎛ ⎞ a1,1c1 + a1,2c2 + ···+ a1,ncn c1 ⎜ a2,1 a2,2 ⎜ ... a2,n ⎟ ⎜ a2,1c1 ⎟ ⎜ . ⎜ . ⎝ . ⎟ ⎝ . ⎟ ⎜ + a2,2c2 + ···+ a2,ncn ⎟ . ⎠ . ⎠ = ⎜ . ⎝ . ⎟ . ⎠ cn am,1 am,2 ... am,n am,1c1 + am,2c2 + ···+ am,ncn D
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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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28 Chapter 1. Linear Systems 3.13 E
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30 Chapter 1. Linear Systems (a) 2x
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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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56 Chapter 1. Linear Systems First,
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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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128 Chapter 2. Vector Spaces Anothe
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130 Chapter 2. Vector Spaces (a)
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136 Chapter 2. Vector Spaces 4.12 E
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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 303 (b) Prove
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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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Chapter 5 Similarity While studying
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Section III. Nilpotence 369 Proof.
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Section IV. Jordan Form 379 5.IV Jo
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Section IV. Jordan Form 385 � 1.1
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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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