Linear Algebra

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$MATLAB C++ Math Library Reference$

212 Chapter 3. Maps Between Spaces Representing a scalar multiple of a map works the same way. 1.2 Example If t is a transformation represented by � 1 RepB,D(t) = 1 � 0 1 so that � � v1 �v = � ↦→ v1 B,D then the scalar multiple map 5t acts in this way. � � v1 �v = v2 ↦−→ � � 5v1 5v1 +5v2 Therefore, this is the matrix representing 5t. � � 5 0 RepB,D(5t) = 5 5 B v2 D B B,D =5· t(�v) v1 + v2 � D = t(�v) 1.3 Definition The sum of two same-sized matrices is their entry-by-entry sum. The scalar multiple of a matrix is the result of entry-by-entry scalar multiplication. 1.4 Remark These extend the vector addition and scalar multiplication operations that we defined in the first chapter. 1.5 Theorem Let h, g : V → W be linear maps represented with respect to bases B,D by the matrices H and G, and let r be a scalar. Then the map h + g : V → W is represented with respect to B,D by H + G, and the map r · h: V → W is represented with respect to B,D by rH. Proof. Exercise 8; generalize the examples above. QED A notable special case of scalar multiplication is multiplication by zero. For any map 0 · h is the zero homomorphism and for any matrix 0 · H is the zero matrix. 1.6 Example The zero map from any three-dimensional space to any twodimensional space is represented by the 2×3 zero matrix � � 0 0 0 Z = 0 0 0 no matter which domain and codomain bases are used. Exercises � 1.7 Perform � the indicated � � operations, � if defined. 5 −1 2 2 1 4 (a) + 6 1 1 3 0 5 � � 2 −1 −1 (b) 6 · 1 2 3
Section IV. Matrix Operations 213 � � � � 2 1 2 1 (c) + 0 3 0 3 � � � � 1 2 −1 4 (d) 4 +5 3 −1 −2 1 � � � � 2 1 1 1 4 (e) 3 +2 3 0 3 0 5 1.8 Prove Theorem 1.5. (a) Prove that matrix addition represents addition of linear maps. (b) Prove that matrix scalar multiplication represents scalar multiplication of linear maps. � 1.9 Prove each, where the operations are defined, where G, H, andJare matrices, where Z is the zero matrix, and where r and s are scalars. (a) Matrix addition is commutative G + H = H + G. (b) Matrix addition is associative G +(H + J) =(G + H)+J. (c) The zero matrix is an additive identity G + Z = G. (d) 0 · G = Z (e) (r + s)G = rG + sG (f) Matrices have an additive inverse G +(−1) · G = Z. (g) r(G + H) =rG + rH (h) (rs)G = r(sG) 1.10 Fix domain and codomain spaces. In general, one matrix can represent many different maps with respect to different bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices? � 1.11 Let V and W be vector spaces of dimensions n and m. Show that the space L(V,W) of linear maps from V to W is isomorphic to Mm×n. � 1.12 Show that it follows from the prior questions that for any six transformations t1,...,t6 : R 2 → R 2 there are scalars c1,...,c6 ∈ R such that c1t1 + ···+ c6t6 is the zero map. (Hint: this is a bit of a misleading question.) 1.13 The trace of a square matrix is the sum of the entries on the main diagonal (the 1, 1 entry plus the 2, 2 entry, etc.; we will see the significance of the trace in Chapter Five). Show that trace(H + G) =trace(H)+trace(G). Is there a similar result for scalar multiplication? 1.14 Recall that the transpose of a matrix M is another matrix, whose i, j entry is the j, i entry of M. Verifiy these identities. (a) (G + H) trans = G trans + H trans (b) (r · H) trans = r · H trans � 1.15 A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if the matrix equals its transpose. (a) Prove that for any H, the matrix H + H trans is symmetric. Does every symmetric matrix have this form? (b) Prove that the set of n×n symmetric matrices is a subspace of Mn×n. � 1.16 (a) How does matrix rank interact with scalar multiplication—can a scalar product of a rank n matrix have rank less than n? Greater? (b) How does matrix rank interact with matrix addition—can a sum of rank n matrices have rank less than n? Greater?
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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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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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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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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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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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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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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 I. Definition 311 � 3.23
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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 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 III. Nilpotence 373 Now, wi
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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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