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340 Sample Second Midterm 340
F Sample Final Exam Here are some worked problems typical for what you might expect on a final examination. 1. Define the following terms: (a) An orthogonal matrix. (b) A basis for a vector space. (c) The span of a set of vectors. (d) The dimension of a vector space. (e) An eigenvector. (f) A subspace of a vector space. (g) The kernel of a linear transformation. (h) The nullity of a linear transformation. (i) The image of a linear transformation. (j) The rank of a linear transformation. (k) The characteristic polynomial of a square matrix. (l) An equivalence relation. (m) A homogeneous solution to a linear system of equations. (n) A particular solution to a linear system of equations. (o) The general solution to a linear system of equations. (p) The direct sum of a pair of subspaces of a vector space. 341
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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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Page 293 and 294: 16.2 Image 293 Example 148 of calcu
Page 295 and 296: 16.2 Image 295 Thus ⎧⎛ ⎞ ⎛
Page 297 and 298: 16.3 Summary 297 Reading homework:
Page 299 and 300: 16.4 Review Problems 299 16.4 Revie
Page 301 and 302: 16.4 Review Problems 301 Find the d
Page 303 and 304: 17 Least squares and Singular Value
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Page 307 and 308: 17.1 Projection Matrices 307 MX = V
Page 309 and 310: 17.2 Singular Value Decomposition 3
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Page 313 and 314: 17.3 Review Problems 313 • v T M
Page 315 and 316: A List of Symbols ∈ ∼ R I n Pn
Page 317 and 318: B Fields Definition A field F is a
Page 319 and 320: Online Resources C Here are some in
Page 321 and 322: D Sample First Midterm Here are som
Page 323 and 324: 323 8. What are the four main thing
Page 325 and 326: 325 2. (a) The augmented matrix ⎛
Page 327 and 328: 327 Since M T M −1 ≠ I, it foll
Page 329 and 330: 329 (b) This is a vector space. Alt
Page 331 and 332: E Sample Second Midterm Here are so
Page 333 and 334: 333 6. Suppose λ is an eigenvalue
Page 335 and 336: 335 Thus ⎛ ⎞ ⎛ ⎞ ⎛ 3 1
Page 337 and 338: 337 8. The associated eigenvalues s
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Page 343 and 344: 343 (h) Is M T M diagonalizable? (i
Page 345 and 346: 345 after one orbit the satellite w
Page 347 and 348: 347 (e) Is there a direction in whi
Page 349 and 350: 349 well into the future. Then F 3
Page 351 and 352: 351 Solutions (a) Write down a line
Page 353 and 354: 353 Now solve UX = W by back substi
Page 355 and 356: 355 Now zero out the top row by sub
Page 357 and 358: 357 Finally, for λ = 3 2 ⎛ − 3
Page 359 and 360: 359 11. (a) d2 X dt 2 (b) Since thi
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Page 365 and 366: 365 (b) The rows of M span (kerL)
Page 367 and 368: G Movie Scripts G.1 What is Linear
Page 369 and 370: G.2 Systems of Linear Equations 369
Page 377 and 378: G.3 Vectors in Space n-Vectors 377
Page 379 and 380: G.4 Vector Spaces 379 idea is to pl
Page 381 and 382: G.4 Vector Spaces 381 We are half-w
Page 383 and 384: G.5 Linear Transformations 383 You
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
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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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