Linear Algebra

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$MATLAB C++ Math Library Reference$

106 Chapter 2. Vector Spaces Looking for a linear relationship ⎛ c1 ⎝ 1 ⎞ ⎛ 0⎠ + c2 ⎝ 0 0 ⎞ ⎛ 2⎠ + c3 ⎝ 0 1 ⎞ ⎛ 2⎠ + c4 ⎝ 0 0 ⎞ ⎛ −1⎠ + c5 ⎝ 1 3 ⎞ ⎛ 3⎠ = ⎝ 0 0 ⎞ 0⎠ 0 gives a three equations/five unknowns linear system whose solution set can be paramatrized in this way. ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ c1 −1 −3 ⎜c2⎟ ⎜ ⎜ ⎟ ⎜−1⎟ ⎜ ⎟ ⎜−3/2 ⎟ � { ⎜c3⎟ ⎜ ⎟ = c3 ⎜ 1 ⎟ + c5 ⎜ 0 ⎟ � ⎟ c3,c5 ∈ R} ⎝c4⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ 0 1 c5 Setting, say, c3 =0andc5 = 1 shows that the fifth vector is a linear combination of the first two. Thus, Lemma 1.1 gives that this set ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 1 0 S1 = { ⎝0⎠ , ⎝2⎠ , ⎝2⎠ , ⎝−1⎠} 0 0 0 1 has the same span as S0. Similarly, the third vector of the new set S1 is a linear combination of the first two and we get ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 S2 = { ⎝0⎠ , ⎝2⎠ , ⎝−1⎠} 0 0 1 with the same span as S1 and S0, but with one difference. This last set is linearly independent (this is easily checked), and so removal of any of its vectors will shrink the span. We finish this subsection by recasting that example as a theorem that any finite spanning set has a subset with the same span that is linearly independent. To prove that result we will first need some facts about how linear independence and dependence, which are properties of sets, interact with the subset relation between sets. 1.13 Lemma Any subset of a linearly independent set is also linearly independent. Any superset of a linearly dependent set is also linearly dependent. Proof. This is clear. QED Restated, independence is preserved by subset and dependence is preserved by superset. Those are two of the four possible cases of interaction that we can consider. The third case, whether linear dependence is preserved by the subset operation, is covered by Example 1.12, which gives a linearly dependent set S0 with a subset S1 that is linearly dependent and another subset S2 that is linearly independent. That leaves one case, whether linear independence is preserved by superset. The next example shows what can happen.
Section II. <strong>Linear</strong> Independence 107 1.14 Example Here are some linearly independent sets from R3 and their supersets. ⎛ (1) If S1 = { ⎝ 1 ⎞ 0⎠} then the span [S1] isthex-axis. 0 A linearly dependent superset: ⎛ { ⎝ 1 ⎞ ⎛ 0⎠ , ⎝ 0 −3 ⎞ 0 ⎠} A linearly independent superset: ⎛ 0 { ⎝ 1 ⎞ ⎛ 0⎠ , ⎝ 0 0 ⎞ 1⎠} 0 ⎛ ⎞ ⎛ ⎞ 1 0 (2) If S2 = { ⎝0⎠ , ⎝1⎠} then [S2] isthexy-plane. 0 0 ⎛ ⎞ 1 ⎛ ⎞ 0 ⎛ ⎞ 3 A linearly dependent superset: { ⎝0⎠ , ⎝1⎠ , ⎝−2⎠} ⎛ 0 ⎞ 1 ⎛ 0 ⎞ 0 ⎛ 0 ⎞ 0 A linearly independent superset: { ⎝0⎠ , ⎝1⎠ , ⎝0⎠} 0 0 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 (3) If S3 = { ⎝0⎠ , ⎝1⎠ , ⎝0⎠} then [S3] is all of R 0 0 1 3 . ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 2 A linearly dependent superset: { ⎝0⎠ , ⎝1⎠ , ⎝0⎠ , ⎝−1⎠} 0 0 1 3 There are no linearly independent supersets. (Checking the dependence or independence of these sets is easy.) So in general a linearly independent set can have some supersets that are dependent and some supersets that are independent. We can characterize when a superset of a independent set is dependent and when it is independent. 1.15 Lemma Where S is a linearly independent subset of a vector space V , S ∪{�v} is linearly dependent if and only if �v ∈ [S] for any �v ∈ V with �v �∈ S.
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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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12 Chapter 1. Linear Systems can al
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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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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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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 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: Projective Geometry 337 Topi
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Topic: Projective Geometry 341 of t
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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 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 397 � 5 4
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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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