Linear Algebra

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$MATLAB C++ Math Library Reference$

204 Chapter 3. Maps Between Spaces (b) Repeat the prior item, only rotate about the y-axis instead. (Put the person’s head at �e2.) (c) Repeat, about the z-axis. (d) Extend the prior item to R 4 .(Hint: ‘rotate about the z-axis’ can be restated as ‘rotate parallel to the xy-plane’.) 1.30 (Schur’s Triangularization Lemma) (a) Let U be a subspace of V and fix bases BU ⊆ BV . What is the relationship between the representation of a vector from U with respect to BU and the representation of that vector (viewed as a member of V )withrespecttoBV ? (b) What about maps? (c) Fix a basis B = 〈 � β1,..., � βn〉 for V and observe that the spans [{�0}] ={�0} ⊂[{ � β1}] ⊂ [{ � β1, � β2}] ⊂ ··· ⊂[B] =V form a strictly increasing chain of subspaces. Show that for any linear map h: V → W there is a chain W0 = {�0} ⊆W1 ⊆ ··· ⊆ Wm = W of subspaces of W such that h([{ � β1,..., � βi}]) ⊂ Wi for each i. (d) Conclude that for every linear map h: V → W there are bases B,D so the matrix representing h with respect to B,D is upper-triangular (that is, each entry hi,j with i>jis zero). (e) Is an upper-triangular representation unique? 3.III.2 Any Matrix Represents a <strong>Linear</strong> Map The prior subsection shows that the action of a linear map h is described by a matrix H, with respect to appropriate bases, in this way. ⎛ ⎞ ⎛ ⎞ �v = ⎜ ⎝ v1 . vn ⎟ ⎠ B h ↦−→ H ⎜ ⎝ h1,1v1 + ···+ h1,nvn . hm,1v1 + ···+ hm,nvn ⎟ ⎠ D = h(�v) In this subsection, we will show the converse, that each matrix represents a linear map. Recall that, in the definition of the matrix representation of a linear map, the number of columns of the matrix is the dimension of the map’s domain and the number of rows of the matrix is the dimension of the map’s codomain. Thus, for instance, a 2×3 matrix cannot represent a map from R 5 to R 4 . The next result says that, beyond this restriction on the dimensions, there are no other limitations: the 2×3 matrix represents a map from any three-dimensional space to any two-dimensional space. 2.1 Theorem Any matrix represents a homomorphism between vector spaces of appropriate dimensions, with respect to any pair of bases.
Section III. Computing <strong>Linear</strong> Maps 205 Proof. For the matrix ⎛ ⎜ H = ⎜ ⎝ h1,1 h1,2 ... h1,n h2,1 h2,2 ... h2,n . hm,1 hm,2 ... hm,n fix any n-dimensional domain space V and any m-dimensional codomain space W . Also fix bases B = 〈 � β1,..., � βn〉 and D = 〈 �δ1,..., �δm〉 for those spaces. Define a function h: V → W by: where �v in the domain is represented as ⎛ ⎞ Rep B(�v) = then its image h(�v) is the member the codomain represented by ⎛ ⎞ h1,1v1 + ···+ h1,nvn ⎜ RepD( h(�v))= . ⎝ . ⎟ . ⎠ hm,1v1 + ···+ hm,nvn that is, h(�v) =h(v1 � β1 + ···+ vn � βn) is defined to be (h1,1v1 + ···+ h1,nvn) · �δ1 + ···+(hm,1v1 + ···+ hm,nvn) · �δm. (This is well-defined by the uniqueness of the representation RepB(�v).) Observe that h has simply been defined to make it the map that is represented with respect to B,D by the matrix H. So to finish, we need only check that h is linear. If �v,�u ∈ V are such that ⎛ ⎞ ⎛ ⎞ Rep B(�v) = ⎜ ⎝ v1 . vn and c, d ∈ R then the calculation ⎜ ⎝ v1 . vn ⎟ ⎠ B ⎟ ⎠ and Rep B(�u) = h(c�v + d�u) = � h1,1(cv1 + du1)+···+ h1,n(cvn + dun) � · �δ1 + ···+ � hm,1(cv1 + du1)+···+ hm,n(cvn + dun) � · �δm = c · h(�v)+d · h(�u) provides this verification. QED 2.2 Example Which map the matrix represents depends on which bases are used. If � � � � � � � � � � 1 0 1 0 0 1 H = , B1 = D1 = 〈 , 〉, and B2 = D2 = 〈 , 〉, 0 0 0 1 1 0 ⎞ ⎟ ⎠ ⎜ ⎝ u1 . un 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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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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14 Chapter 1. Linear Systems Matric
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16 Chapter 1. Linear Systems Scalar
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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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56 Chapter 1. Linear Systems First,
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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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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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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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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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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 II. Geometry of Determinant
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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 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 361 Proof. A
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Section III. Nilpotence 365 5.III N
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Section III. Nilpotence 369 Proof.
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Section III. Nilpotence 371 The new
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Section IV. Jordan Form 379 5.IV Jo
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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 393 2.13 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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