Linear Algebra

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$MATLAB C++ Math Library Reference$

222 Chapter 3. Maps Between Spaces 3.1 Example Multiplying by an appropriately-sized zero matrix from the left or from the right � �� 0 0 1 3 2 = 0 0 −1 1 −1 results in a zero matrix. � 0 0 0 0 0 0 � � �� 2 3 0 0 = 1 4 0 0 � � 0 0 0 0 After zero matrices, the matrices whose actions are easiest to understand are the ones with a single nonzero entry. 3.2 Definition A matrix with all zeroes except for a one in the i, j entry is an i, j unit matrix. 3.3 Example This is the 1, 2 unit matrix with three rows and two columns, multiplying from the left. ⎛ ⎞ ⎛ ⎞ 0 1 � � 7 8 ⎝0 0⎠ 5 6 = ⎝0 0⎠ 7 8 0 0 0 0 Acting from the left, an i, j unit matrix copies row j of the multiplicand into row i of the result. From the right an i, j unit matrix copies column i of the multiplicand into column j of the result. ⎛ 1 ⎝4 2 5 ⎞ ⎛ 3 0 6⎠⎝0 ⎞ ⎛ 1 0 0⎠ = ⎝0 ⎞ 1 4⎠ 7 8 9 0 0 0 7 3.4 Example Rescaling these matrices simply rescales the result. This is the action from the left of the matrix that is twice the one in the prior example. ⎛ ⎞ ⎛ ⎞ 0 2 � � 14 16 ⎝0 0⎠ 5 6 = ⎝ 0 0⎠ 7 8 0 0 0 0 And this is the action of the matrix that is minus three times the one from the prior example. ⎛ 1 ⎝4 2 5 ⎞ ⎛ 3 0 6⎠⎝0 ⎞ ⎛ −3 0 0 ⎠ = ⎝0 ⎞ −3 −12⎠ 7 8 9 0 0 0 −21 Next in complication are matrices with two nonzero entries. There are two cases. If a left-multiplier has entries in different rows then their actions don’t interact.
Section IV. Matrix Operations 223 3.5 Example ⎛ 1 ⎝0 0 0 ⎞ ⎛ 0 1 2⎠⎝4 2 5 ⎞ ⎛ 3 1 6⎠ =( ⎝0 0 0 ⎞ ⎛ 0 0 0⎠ + ⎝0 0 0 ⎞ ⎛ 0 1 2⎠) ⎝4 2 5 ⎞ 3 6⎠ 0 0 0 7 8 9 0 ⎛ 1 = ⎝0 0 2 0 0 0 ⎞ ⎛ 3 0 0⎠ + ⎝14 0 0 7 ⎞ 0 0 16 18⎠ 8 9 0 ⎛ 1 = ⎝14 0 0 ⎞ 2 3 16 18⎠ 0 0 0 0 0 0 But if the left-multiplier’s nonzero entries are in the same row then that row of the result is a combination. 3.6 Example ⎛ 1 ⎝0 0 0 ⎞ ⎛ 2 1 0⎠⎝4 2 5 ⎞ ⎛ 3 1 6⎠ =( ⎝0 0 0 ⎞ ⎛ 0 0 0⎠ + ⎝0 0 0 ⎞ ⎛ 2 1 0⎠) ⎝4 2 5 ⎞ 3 6⎠ 0 0 0 7 8 9 0 ⎛ 1 = ⎝0 0 2 0 0 0 ⎞ ⎛ 3 14 0⎠ + ⎝ 0 0 0 7 ⎞ 16 18 0 0⎠ 8 9 0 ⎛ 15 0 0 ⎞ 18 21 0 0 0 = ⎝ 0 0 0⎠ 0 0 0 Right-multiplication acts in the same way, with columns. These observations about matrices that are mostly zeroes extend to arbitrary matrices. 3.7 Lemma In a product of two matrices G and H, the columns of GH are formed by taking G times the columns of H ⎛ ⎞ . ⎜ . . ⎟ G · ⎜� ⎝h1 ··· �hn ⎟ ⎠ . . . . = ⎛ . ⎜ . . ⎜ ⎝G · �h1 ··· G · � ⎞ ⎟ hn ⎟ ⎠ . . . . and the rows of GH are formed by taking the rows of G times H ⎛ ⎞ ⎛ ⎞ ··· �g1 ··· ··· �g1 · H ··· ⎜ ⎟ ⎜ ⎟ ⎜ . ⎝ . ⎟ ⎠ · H = ⎜ . ⎝ . ⎟ ⎠ ··· �gr ··· (ignoring the extra parentheses). ··· �gr · H ···
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� � � �1 2� � �3 1�
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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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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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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 309 read alou
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Section I. Definition 311 � 3.23
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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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