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100 Vectors in Space, n-Vectors 100
Vector Spaces 5 As suggested at the end of chapter 4, the vector spaces R n are not the only vector spaces. We now give a general definition that includes R n for all values of n, and R S for all sets S, and more. This mathematical structure is applicable to a wide range of real-world problems and allows for tremendous economy of thought; the idea of a basis for a vector space will drive home the main idea of vector spaces; they are sets with very simple structure. The two key properties of vectors are that they can be added together and multiplied by scalars. Thus, before giving a rigorous definition of vector spaces, we restate the main idea. A vector space is a set that is closed under addition and scalar multiplication. Definition A vector space (V, +, . , R) is a set V with two operations + and · satisfying the following properties for all u, v ∈ V and c, d ∈ R: (+i) (Additive Closure) u + v ∈ V . Adding two vectors gives a vector. (+ii) (Additive Commutativity) u + v = v + u. Order of addition does not matter. (+iii) (Additive Associativity) (u + v) + w = u + (v + w). Order of adding many vectors does not matter. 101
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Linear Algebra David Cherney, Tom D
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Contents 1 What is Linear Algebra?
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5 7.3 Properties of Matrices . . .
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7 16 Kernel, Range, Nullity, Rank 2
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What is Linear Algebra? 1 Many diff
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1.1 Organizing Information 11 ⎛
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1.2 What are Vectors? 13 (C) Polyno
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1.3 What are Linear Functions? 15 1
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1.3 What are Linear Functions? 17 W
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1.3 What are Linear Functions? 19 F
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1.4 So, What is a Matrix? 21 Each b
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1.4 So, What is a Matrix? 23 This i
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1.4 So, What is a Matrix? 25 We hav
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1.4 So, What is a Matrix? 27 = (2ax
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1.4 So, What is a Matrix? 29 Exampl
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1.5 Review Problems 31 Remember tha
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1.5 Review Problems 33 6. Matrix Mu
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1.5 Review Problems 35 (iv) Your an
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Systems of Linear Equations 2 2.1 G
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2.1 Gaussian Elimination 39 Entries
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2.1 Gaussian Elimination 41 called
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2.1 Gaussian Elimination 43 Algorit
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2.1 Gaussian Elimination 45 Advance
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2.1 Gaussian Elimination 47 ⎧ ⎪
Page 49 and 50: 2.2 Review Problems 49 2. Solve the
Page 51 and 52: 2.2 Review Problems 51 8. Show that
Page 53 and 54: 2.3 Elementary Row Operations 53 Ex
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Page 57 and 58: 2.3 Elementary Row Operations 57
Page 59 and 60: 2.3 Elementary Row Operations 59 wh
Page 61 and 62: 2.4 Review Problems 61 The LDU fact
Page 63 and 64: 2.5 Solution Sets for Systems of Li
Page 69 and 70: 2.6 Review Problems 69 so that a 2
Page 71 and 72: The Simplex Method 3 In Chapter 2,
Page 73 and 74: 3.2 Graphical Solutions 73 3.2 Grap
Page 75 and 76: 3.3 Dantzig’s Algorithm 75 Here w
Page 77 and 78: 3.3 Dantzig’s Algorithm 77 ⎛
Page 79 and 80: 3.4 Pablo Meets Dantzig 79 These ar
Page 81 and 82: 3.5 Review Problems 81 their profit
Page 83 and 84: Vectors in Space, n-Vectors 4 To co
Page 85 and 86: 4.2 Hyperplanes 85 4.2 Hyperplanes
Page 87 and 88: 4.2 Hyperplanes 87 unless any of th
Page 89 and 90: 4.3 Directions and Magnitudes 89 Th
Page 91 and 92: 4.3 Directions and Magnitudes 91 Ex
Page 93 and 94: 4.3 Directions and Magnitudes 93 Th
Page 95 and 96: 4.4 Vectors, Lists and Functions: R
Page 97 and 98: 4.5 Review Problems 97 4.5 Review P
Page 99: 4.5 Review Problems 99 where the ve
Page 103 and 104: 5.1 Examples of Vector Spaces 103 E
Page 105 and 106: 5.1 Examples of Vector Spaces 105 E
Page 107 and 108: 5.2 Other Fields 107 {( ∣∣∣ }
Page 109 and 110: 5.3 Review Problems 109 5.3 Review
Page 111 and 112: Linear Transformations 6 The main o
Page 113 and 114: 6.1 The Consequence of Linearity 11
Page 115 and 116: 6.3 Linear Differential Operators 1
Page 117 and 118: 6.4 Bases (Take 1) 117 sum of multi
Page 119 and 120: 6.5 Review Problems 119 2. If f is
Page 121 and 122: Matrices 7 Matrices are a powerful
Page 123 and 124: 7.1 Linear Transformations and Matr
Page 129 and 130: 7.2 Review Problems 129 Linear oper
Page 131 and 132: 7.2 Review Problems 131 (c) Find th
Page 133 and 134: 7.3 Properties of Matrices 133 (m)
Page 135 and 136: 7.3 Properties of Matrices 135 Exam
Page 137 and 138: 7.3 Properties of Matrices 137 This
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Page 141 and 142: 7.3 Properties of Matrices 141 Some
Page 143 and 144: 7.3 Properties of Matrices 143 •
Page 145 and 146: 7.3 Properties of Matrices 145 7.3.
Page 147 and 148: 7.4 Review Problems 147 ⎛ ⎞ ⎛
Page 149 and 150: 7.4 Review Problems 149 Give a few
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7.5 Inverse Matrix 151 Figure 7.1:
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7.5 Inverse Matrix 153 At this poin
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7.6 Review Problems 155 Notice that
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7.6 Review Problems 157 (a) Compute
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7.7 LU Redux 159 7.7 LU Redux Certa
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7.7 LU Redux 161 ⎛ ⎞ u • Step
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7.7 LU Redux 163 so we would like t
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7.7 LU Redux 165 Reading homework:
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7.8 Review Problems 167 (k) Try to
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Determinants 8 Given a square matri
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8.1 The Determinant Formula 171 We
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8.1 The Determinant Formula 173 Thi
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8.2 Elementary Matrices and Determi
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8.3 Review Problems 183 Use row ope
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8.3 Review Problems 185 express you
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8.4 Properties of the Determinant 1
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8.5 Review Problems 193 Figure 8.8:
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Subspaces and Spanning Sets 9 It is
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9.2 Building Subspaces 197 Note tha
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9.2 Building Subspaces 199 of the f
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9.2 Building Subspaces 201 Hence, t
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10 Linear Independence Consider a p
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10.1 Showing Linear Dependence 205
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10.2 Showing Linear Independence 20
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10.3 From Dependent Independent 209
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10.4 Review Problems 211 (b) If pos
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11 Basis and Dimension In chapter 1
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215 Worked Example Next, we would l
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11.1 Bases in R n . 217 a basis for
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11.2 Matrix of a Linear Transformat
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11.3 Review Problems 221 Alternativ
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11.3 Review Problems 223 6. Let S n
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12 Eigenvalues and Eigenvectors In
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12.1 Invariant Directions 227 as we
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12.1 Invariant Directions 229 Figur
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12.1 Invariant Directions 231 Readi
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12.2 The Eigenvalue-Eigenvector Equ
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12.2 The Eigenvalue-Eigenvector Equ
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12.3 Eigenspaces 237 is also an eig
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12.4 Review Problems 239 4. Let L b
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13 Diagonalization Given a linear t
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13.2 Change of Basis 243 Here, the
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13.2 Change of Basis 245 Meanwhile,
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13.3 Changing to a Basis of Eigenve
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13.4 Review Problems 249 (t n , . .
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13.4 Review Problems 251 (a) Show t
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14 Orthonormal Bases and Complement
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14.2 Orthogonal and Orthonormal Bas
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14.2 Orthogonal and Orthonormal Bas
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14.3 Relating Orthonormal Bases 259
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14.4 Gram-Schmidt & Orthogonal Comp
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14.4 Gram-Schmidt & Orthogonal Comp
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14.5 QR Decomposition 265 First, we
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14.6 Orthogonal Complements 267 vec
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14.6 Orthogonal Complements 269 The
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14.6 Orthogonal Complements 271 Exa
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14.7 Review Problems 273 (c) Suppos
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14.7 Review Problems 275 7. (a) Sho
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15 Diagonalizing Symmetric Matrices
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279 Example 141 The matrix M = ( )
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15.1 Review Problems 281 To diagona
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15.1 Review Problems 283 (b) Explai
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16 Kernel, Range, Nullity, Rank Giv
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16.2 Image 287 It might occur to yo
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16.2 Image 289 Figure 16.1: For the
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16.2 Image 291 Theorem 16.2.1. A fu
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16.2 Image 293 Example 148 of calcu
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16.2 Image 295 Thus ⎧⎛ ⎞ ⎛
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16.3 Summary 297 Reading homework:
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16.4 Review Problems 299 16.4 Revie
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16.4 Review Problems 301 Find the d
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17 Least squares and Singular Value
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305 However, the converse is often
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17.1 Projection Matrices 307 MX = V
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17.2 Singular Value Decomposition 3
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17.2 Singular Value Decomposition 3
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17.3 Review Problems 313 • v T M
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A List of Symbols ∈ ∼ R I n Pn
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B Fields Definition A field F is a
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Online Resources C Here are some in
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D Sample First Midterm Here are som
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323 8. What are the four main thing
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325 2. (a) The augmented matrix ⎛
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327 Since M T M −1 ≠ I, it foll
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329 (b) This is a vector space. Alt
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E Sample Second Midterm Here are so
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333 6. Suppose λ is an eigenvalue
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335 Thus ⎛ ⎞ ⎛ ⎞ ⎛ 3 1
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337 8. The associated eigenvalues s
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339 10. (i) We say that the vectors
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F Sample Final Exam Here are some w
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343 (h) Is M T M diagonalizable? (i
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345 after one orbit the satellite w
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347 (e) Is there a direction in whi
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349 well into the future. Then F 3
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351 Solutions (a) Write down a line
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353 Now solve UX = W by back substi
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355 Now zero out the top row by sub
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357 Finally, for λ = 3 2 ⎛ − 3
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359 11. (a) d2 X dt 2 (b) Since thi
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361 (b) Yes, the matrix M is symmet
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363 and since ker L = {0 V } we hav
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365 (b) The rows of M span (kerL)
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G Movie Scripts G.1 What is Linear
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G.2 Systems of Linear Equations 369
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G.3 Vectors in Space n-Vectors 377
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G.4 Vector Spaces 379 idea is to pl
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G.4 Vector Spaces 381 We are half-w
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G.5 Linear Transformations 383 You
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G.6 Matrices 385 G.6 Matrices Adjac
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G.6 Matrices 387 so this pair of ma
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G.6 Matrices 389 Hint for Review Qu
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G.6 Matrices 391 So this means that
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G.6 Matrices 393 Using an LU Decomp
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G.7 Determinants 395 we could fit t
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G.7 Determinants 397 2. As a matrix
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G.7 Determinants 399 ⎛ 1 Sj(λ) i
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G.7 Determinants 401 Now note that
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G.8 Subspaces and Spanning Sets 403
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G.9 Linear Independence 405 notion
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G.10 Basis and Dimension 407 G.10 B
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G.11 Eigenvalues and Eigenvectors 4
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G.12 Diagonalization 415 G.12 Diago
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G.12 Diagonalization 417 fruit (s,
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G.12 Diagonalization 419 Note that
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G.13 Orthonormal Bases and Compleme
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G.14 Diagonalizing Symmetric Matric
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G.15 Kernel, Range, Nullity, Rank 4
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Index Action, 389 Angle between vec
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INDEX 435 Linearly independent, 204
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