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432 Movie Scripts G.16 Least Squares and Singular Values Least Squares: Hint for Review Problem 1 Lets work through this problem. Let L : U → V be a linear transformation. Suppose v ∈ L(U) and you have found a vector u ps that obeys L(u ps ) = v. Explain why you need to compute ker L to describe the solution space of the linear system L(u) = v. Remember the property of linearity that comes along with any linear transformation: L(ax + by) = aL(x) + bL(y) for scalars a and b. This allows us to break apart and recombine terms inside the transformation. Now suppose we have a solution x where L(x) = v. If we have an vector y ∈ ker L then we know L(y) = 0. If we add the equations together L(x) + L(y) = L(x + y) = v + 0 we get another solution for free. Now we have two solutions, is that all? Hint for Review Problem 2 For the first part, what is the transpose of a 1 × 1 matrix? For the other two parts, note that v v = v T v. Can you express this in terms of ‖v‖? Also you need the trivial kernel only for the last part and just think about the null space of M. It might help to substitute w = Mx. 432
Index Action, 389 Angle between vectors, 90 Anti-symmetric matrix, 149 Back substitution, 160 Base field, 107 Basis, 213 concept of, 195 example of, 209 basis, 117, 118 Bit matrices, 154 Bit Matrix, 155 Block matrix, 142 Calculus Superhero, 305 Canonical basis, see also Standard basis, 408 Captain Conundrum, 97, 305 Cauchy–Schwarz inequality, 92 Change of basis, 242 Change of basis matrix, 243 Characteristic polynomial, 185, 232, 234 Closure, 197 additive, 101 multiplicative, 102 Codomain, 34, 289 Cofactor, 190 Column Space concept of, 23, 138 Column space, 295 Column vector, 134 of a vector, 126 Components of a vector, 127 Composition of functions, 26 Conic sections, 347 Conjugation, 247 Cramer’s rule, 192 Cross product, 31 Determinant, 172 2 × 2 matrix, 170 3 × 3 matrix, 170 Diagonal matrix, 138 Diagonalizable, 242 Diagonalization, 241 concept of, 230 Dimension, 213 433
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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 ⎧ ⎪
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2.2 Review Problems 49 2. Solve the
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2.2 Review Problems 51 8. Show that
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2.3 Elementary Row Operations 53 Ex
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2.3 Elementary Row Operations 55 Ex
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2.3 Elementary Row Operations 57
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2.3 Elementary Row Operations 59 wh
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2.4 Review Problems 61 The LDU fact
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2.5 Solution Sets for Systems of Li
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2.6 Review Problems 69 so that a 2
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The Simplex Method 3 In Chapter 2,
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3.2 Graphical Solutions 73 3.2 Grap
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3.3 Dantzig’s Algorithm 75 Here w
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3.3 Dantzig’s Algorithm 77 ⎛
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3.4 Pablo Meets Dantzig 79 These ar
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3.5 Review Problems 81 their profit
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Vectors in Space, n-Vectors 4 To co
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4.2 Hyperplanes 85 4.2 Hyperplanes
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4.2 Hyperplanes 87 unless any of th
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4.3 Directions and Magnitudes 89 Th
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4.3 Directions and Magnitudes 91 Ex
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4.3 Directions and Magnitudes 93 Th
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4.4 Vectors, Lists and Functions: R
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4.5 Review Problems 97 4.5 Review P
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4.5 Review Problems 99 where the ve
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Vector Spaces 5 As suggested at the
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5.1 Examples of Vector Spaces 103 E
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5.1 Examples of Vector Spaces 105 E
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5.2 Other Fields 107 {( ∣∣∣ }
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5.3 Review Problems 109 5.3 Review
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Linear Transformations 6 The main o
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6.1 The Consequence of Linearity 11
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6.3 Linear Differential Operators 1
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6.4 Bases (Take 1) 117 sum of multi
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6.5 Review Problems 119 2. If f is
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Matrices 7 Matrices are a powerful
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7.1 Linear Transformations and Matr
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7.2 Review Problems 129 Linear oper
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7.2 Review Problems 131 (c) Find th
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7.3 Properties of Matrices 133 (m)
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7.3 Properties of Matrices 135 Exam
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7.3 Properties of Matrices 137 This
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7.3 Properties of Matrices 139 The
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7.3 Properties of Matrices 141 Some
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7.3 Properties of Matrices 143 •
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7.3 Properties of Matrices 145 7.3.
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7.4 Review Problems 147 ⎛ ⎞ ⎛
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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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Page 385 and 386: G.6 Matrices 385 G.6 Matrices Adjac
Page 387 and 388: G.6 Matrices 387 so this pair of ma
Page 389 and 390: G.6 Matrices 389 Hint for Review Qu
Page 391 and 392: G.6 Matrices 391 So this means that
Page 393 and 394: G.6 Matrices 393 Using an LU Decomp
Page 395 and 396: G.7 Determinants 395 we could fit t
Page 397 and 398: G.7 Determinants 397 2. As a matrix
Page 399 and 400: G.7 Determinants 399 ⎛ 1 Sj(λ) i
Page 401 and 402: G.7 Determinants 401 Now note that
Page 403 and 404: G.8 Subspaces and Spanning Sets 403
Page 405 and 406: G.9 Linear Independence 405 notion
Page 407 and 408: G.10 Basis and Dimension 407 G.10 B
Page 409 and 410: G.11 Eigenvalues and Eigenvectors 4
Page 415 and 416: G.12 Diagonalization 415 G.12 Diago
Page 417 and 418: G.12 Diagonalization 417 fruit (s,
Page 419 and 420: G.12 Diagonalization 419 Note that
Page 421 and 422: G.13 Orthonormal Bases and Compleme
Page 429 and 430: G.14 Diagonalizing Symmetric Matric
Page 431: G.15 Kernel, Range, Nullity, Rank 4
Page 435 and 436: INDEX 435 Linearly independent, 204
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