A First Course in Linear Algebra, 2017a

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252 R n −1 1 2 3 4 5 6 y 4 2 x Regression Line Data Points One could clearly do a least squares fit for curves of the form y = ax 2 + bx + c in the same way. In this case you want to solve as well as possible for a,b, andc the system ⎡ ⎤ x 2 ⎡ ⎤ ⎡ ⎤ 1 x 1 1 a y 1 ⎢ ⎥ ⎣ . . . ⎦⎣ b ⎦ ⎢ ⎥ = ⎣ . ⎦ x 2 n x n 1 c y n and one would use the same technique as above. Many other similar problems are important, including many in higher dimensions and they are all solved the same way. Exercises Exercise 4.11.1 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. ⎡ √ 1 ⎤ ⎡ ⎤ ⎡ 6√ 2 √ 3 1 2√ 2 − ⎣ 1 3√ 1 ⎤ 3√ 3 2 3 − 1 √ √ ⎦, ⎣ √ 0 ⎦, ⎣ 1 3√ 3 ⎦ √ 6 2 3 1 2 2 1 3 3 If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span. Exercise 4.11.2 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 1 −1 ⎣ 2 ⎦, ⎣ 0 ⎦, ⎣ 1 ⎦ −1 1 1 If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span. ♠
4.11. Orthogonality and the Gram Schmidt Process 253 Exercise 4.11.3 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 2 0 ⎣ −1 ⎦, ⎣ 1 ⎦, ⎣ 1 ⎦ 1 −1 1 If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span. Exercise 4.11.4 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 2 1 ⎣ −1 ⎦, ⎣ 1 ⎦, ⎣ 2 ⎦ 1 −1 1 If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span. Exercise 4.11.5 Determine whether the following set of vectors is orthogonal. If it is orthogonal, determine whether it is also orthonormal. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 ⎢ 0 ⎥ ⎣ 0 ⎦ , ⎢ 1 ⎥ ⎣ −1 ⎦ , ⎢ 0 ⎥ ⎣ 0 ⎦ 0 0 1 If the set of vectors is orthogonal but not orthonormal, give an orthonormal set of vectors which has the same span. Exercise 4.11.6 Here are some matrices. Label according to whether they are symmetric, skew symmetric, or orthogonal. (a) ⎡ ⎢ ⎣ 1 0 0 0 √2 1 − √ 1 2 ⎤ ⎥ 1 0 √2 √2 1 ⎦ (b) (c) ⎡ ⎣ ⎡ ⎣ 1 2 −3 2 1 4 −3 4 7 0 −2 −3 2 0 −4 3 4 0 ⎤ ⎦ ⎤ ⎦
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with Open Texts Linear A Algebra wi
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A First Course in Linear Algebra an
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A First Course in Linear Algebra an
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iv Table of Contents 3 Determinants
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vi Table of Contents 7.3.5 The Matr
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Chapter 1 Systems of Equations 1.1
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1.1. Systems of Equations, Geometry
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1.2. Systems of Equations, Algebrai
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54 Matrices Consider the following
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56 Matrices Solution. Notice that b
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58 Matrices Proposition 2.10: Prope
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60 Matrices on the left by a vector
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62 Matrices will satisfy the equati
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64 Matrices Solution. First check i
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66 Matrices Solution. First check i
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68 Matrices Proposition 2.25: Prope
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70 Matrices Definition 2.29: Symmet
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72 Matrices Theorem 2.34: Uniquenes
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74 Matrices Let’s solve the first
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76 Matrices Notice that the left ha
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78 Matrices What if the right side,
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80 Matrices First, consider the fol
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82 Matrices You can see that B is t
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84 Matrices = = ⎡ ⎣ ⎡ ⎣ 1 0
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86 Matrices First: ⎡ ⎣ 0 1 0 1
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88 Matrices Proof. Suppose that A a
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90 Matrices It follows that the red
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92 Matrices (h) DE ⎡ Exercise 2.1
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94 Matrices (a) (A − B) 2 = A 2
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96 Matrices Exercise 2.1.42 Let ⎡
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98 Matrices (a) Find the elementary
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100 Matrices One way to find the LU
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102 Matrices which yields very quic
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104 Matrices By Lemma 2.70, this im
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106 Matrices Exercise 2.2.12 Find t
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108 Determinants The 2×2 determina
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110 Determinants Definition 3.7: Th
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112 Determinants The following prov
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114 Determinants 3.1.3 Properties o
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116 Determinants Example 3.23: Mult
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118 Determinants Solution. Consider
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120 Determinants = ∑a 1,l (cof(B)
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122 Determinants Theorem 3.35: Let
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124 Determinants Here, det(C)=−3d
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126 Determinants (a) minor(A) 11 (b
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128 Determinants Exercise 3.1.13 An
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130 Determinants 3.2.1 A Formula fo
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132 Determinants You can verify tha
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134 Determinants The cofactor matri
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136 Determinants Therefore, ∣ ∣
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138 Determinants After row operatio
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140 Determinants Using the given po
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142 Determinants Does there exist a
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144 R n Hence, every element in R 2
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146 R n components are equal. Preci
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148 R n Theorem 4.6: Properties of
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150 R n 4.3 Geometric Meaning of Ve
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152 R n 4.4 Length of a Vector Outc
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154 R n Solution. We will use the f
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156 R n = √ 1 + 9 + 16 = √ 26 I
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158 R n ⃗u 2⃗v ⃗u + 2⃗v ⃗
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160 R n ⎡ Let ⃗q = ⎣ Then, x
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162 R n Let t = x−2 y−1 3 ,t =
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164 R n Example 4.26: Compute a Dot
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166 R n Notice that this proof was
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168 R n Therefore, the cosine of th
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170 R n 4.7.3 Projections In some a
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172 R n We will conclude this secti
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174 R n Exercise 4.7.11 Find proj
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176 R n Solution. By Definition 4.4
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178 R n From the above scalar equat
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180 R n Definition 4.48: Geometric
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182 R n We will now look at an exam
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184 R n 4.9.1 The Box Product Recal
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186 R n = a ∣ e h ⎡ = det⎣ f
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188 R n 4.10 Spanning, Linear Indep
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190 R n 4.10.2 Linearly Independent
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192 R n Theorem 4.66: Linear Indepe
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194 R n Therefore we can write ⎡
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196 R n Now suppose that⃗u ∉ sp
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198 R n A subspace is simply a set
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200 R n Definition 4.80: Basis of a
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Page 218 and 219: 204 R n ⎡ ⎢ ⎣ 1 2 1 1 3 0 1 3
Page 220 and 221: 206 R n Solution. An easy way to do
Page 222 and 223: 208 R n Since {⃗r 1 ,..., p⃗r j
Page 224 and 225: 210 R n Example 4.101: Rank, Column
Page 226 and 227: 212 R n Definition 4.106: Image of
Page 228 and 229: 214 R n Example 4.110: Null Space o
Page 230 and 231: 216 R n Theorem 4.114: Let A be an
Page 232 and 233: 218 R n Exercise 4.10.7 Here are so
Page 234 and 235: 220 R n Exercise 4.10.17 Are the fo
Page 236 and 237: 222 R n Thse vectors can’t possib
Page 238 and 239: 224 R n ⎧⎡ ⎪⎨ Exercise 4.10
Page 240 and 241: 226 R n Exercise 4.10.52 Consider t
Page 242 and 243: 228 R n Exercise 4.10.68 Find the r
Page 244 and 245: 230 R n Solution. We already verifi
Page 246 and 247: 232 R n Thus to find a correspondin
Page 248 and 249: 234 R n That is: We readily compute
Page 250 and 251: 236 R n We will say that the column
Page 252 and 253: 238 R n To show that {⃗v 1 ,··
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Page 256 and 257: 242 R n = = = ⎡ ⎣ ⎡ ⎣ ⎡
Page 258 and 259: 244 R n In order to find W ⊥ , we
Page 260 and 261: 246 R n Now, we need to write ⃗y
Page 262 and 263: 248 R n Theorem 4.150: Existence of
Page 264 and 265: 250 R n Solution. First, consider w
Page 268 and 269: 254 R n Exercise 4.11.7 For U an or
Page 270 and 271: 256 R n and here is a point: (1,1,2
Page 272 and 273: 258 R n What does addition of vecto
Page 274 and 275: 260 R n 4 3 First we want to know t
Page 276 and 277: 262 R n = −58 Newton meters Note
Page 278 and 279: 264 R n Exercise 4.12.19 A girl dra
Page 280 and 281: 266 Linear Transformations numerica
Page 282 and 283: 268 Linear Transformations Exercise
Page 284 and 285: 270 Linear Transformations In this
Page 286 and 287: 272 Linear Transformations ⎡ ⎣
Page 288 and 289: 274 Linear Transformations ⎡ T
Page 290 and 291: 276 Linear Transformations ⎡ (b)
Page 292 and 293: 278 Linear Transformations The nece
Page 294 and 295: 280 Linear Transformations Then, co
Page 296 and 297: 282 Linear Transformations More gen
Page 298 and 299: 284 Linear Transformations Solution
Page 300 and 301: 286 Linear Transformations Exercise
Page 302 and 303: 288 Linear Transformations Definiti
Page 304 and 305: 290 Linear Transformations This tel
Page 306 and 307: 292 Linear Transformations Example
Page 308 and 309: 294 Linear Transformations 2. T is
Page 310 and 311: 296 Linear Transformations Proposit
Page 312 and 313: 298 Linear Transformations Proof. S
Page 314 and 315: 300 Linear Transformations You can
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302 Linear Transformations Exercise
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304 Linear Transformations for exam
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306 Linear Transformations Example
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308 Linear Transformations Since {T
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310 Linear Transformations Find a b
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312 Linear Transformations Theorem
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314 Linear Transformations and one
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316 Linear Transformations Hence
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318 Linear Transformations system.
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320 Linear Transformations Solution
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322 Linear Transformations Exercise
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324 Complex Numbers Theorem 6.1: Pr
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326 Complex Numbers Example 6.5: Co
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328 Complex Numbers Definition 6.10
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330 Complex Numbers Exercise 6.1.5
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332 Complex Numbers Example 6.14: P
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334 Complex Numbers This requires r
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336 Complex Numbers Therefore, x 3
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338 Complex Numbers Consider the fo
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Chapter 7 Spectral Theory 7.1 Eigen
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7.1. Eigenvalues and Eigenvectors o
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7.2. Diagonalization 355 One eigenv
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7.2. Diagonalization 357 In words,
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7.2. Diagonalization 359 Conversely
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7.2. Diagonalization 361 Theorem 7.
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7.2. Diagonalization 363 7.2.3 Comp
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7.2. Diagonalization 365 Exercise 7
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7.3. Applications of Spectral Theor
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7.4. Orthogonality 395 [ x(0) y(0)
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7.4. Orthogonality 397 Solution. Fi
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7.4. Orthogonality 399 Therefore an
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7.4. Orthogonality 401 In the above
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7.4. Orthogonality 403 Theorem 7.62
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7.4. Orthogonality 405 Theorem 7.66
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7.4. Orthogonality 407 λ 1 = 16: s
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7.4. Orthogonality 409 [ ] [ ] AV1
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7.4. Orthogonality 411 = [ 4 0 0 0
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7.4. Orthogonality 413 by the assum
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7.4. Orthogonality 415 operations)
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7.4. Orthogonality 417 Notice that
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7.4. Orthogonality 419 insignifican
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7.4. Orthogonality 421 Usually it w
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7.4. Orthogonality 423 We now turn
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7.4. Orthogonality 425 Solution. No
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7.4. Orthogonality 427 = 2y 2 1 + 8
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7.4. Orthogonality 429 ⎡ A = ⎢
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7.4. Orthogonality 431 Exercise 7.4
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Chapter 8 Some Curvilinear Coordina
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8.1. Polar Coordinates and Polar Gr
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8.2. Spherical and Cylindrical Coor
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Chapter 9 Vector Spaces 9.1 Algebra
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9.1. Algebraic Considerations 451 E
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9.1. Algebraic Considerations 453 H
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9.1. Algebraic Considerations 455
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9.1. Algebraic Considerations 457 =
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9.2. Spanning Sets 465 Show that, w
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9.2. Spanning Sets 467 Clearlynoval
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9.3. Linear Independence 469 9.3 Li
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9.3. Linear Independence 471 Soluti
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9.3. Linear Independence 473 Exerci
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9.3. Linear Independence 475 Exerci
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9.4. Subspaces and Basis 477 (b) {
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9.4. Subspaces and Basis 479 1. U
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9.4. Subspaces and Basis 481 It fol
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9.4. Subspaces and Basis 483 But th
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9.4. Subspaces and Basis 485 Then {
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9.4. Subspaces and Basis 487 Then f
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9.4. Subspaces and Basis 489 Theref
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9.4. Subspaces and Basis 491 The re
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9.6. Linear Transformations 493 2.
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9.6. Linear Transformations 495 2.
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9.6. Linear Transformations 497 The
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9.7. Isomorphisms 499 9.7 Isomorphi
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9.7. Isomorphisms 501 Example 9.66:
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9.7. Isomorphisms 503 Example 9.71:
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9.7. Isomorphisms 505 Since T is on
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9.7. Isomorphisms 507 for ∑ n i=1
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9.7. Isomorphisms 509 Exercises Exe
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9.7. Isomorphisms 511 for example.
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9.8. The Kernel And Image Of A Line
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9.8. The Kernel And Image Of A Line
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9.9. The Matrix of a Linear Transfo
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Appendix A Some Prerequisite Topics
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A.2. Well Ordering and Induction 53
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A.2. Well Ordering and Induction 53
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Appendix B Selected Exercise Answer
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537 1.2.33 Solution is: [x = 2 −
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539 2.1.7 0A =(0 + 0)A = 0A + 0A.No
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541 [ 2.1.18 Let A = 2.1.20 Let A =
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543 2.1.40 2.1.41 ⎡ ⎣ ⎡ ⎣ 1
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545 2.2.10 An LU factorization of t
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547 3.1.14 If the determinant is no
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549 ⎡ 3.2.2 det⎣ 1 2 0 0 2 1 3
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551 3.2.16 This follows because det
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553 4.7.16 ( ) ( ) ⃗u •⃗v ⃗
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555 It follows that the expression
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557 4.11.6 (a) Orthogonal. (b) Symm
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559 4.11.14 Then a solution is ⎡
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561 4.12.23 ⎡ ⎢ ⃗F 1 • ⎣
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563 5.2.13 ⎡ 1 ⎣ 10 1 0 3 0 0 0
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565 Now to write in terms of (a,b),
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567 ⎡ 5.9.7 Solution is: ⎣ −
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569 6.1.6 If p(z)=0, then you have
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571 7.1.2 Say AX = λX.ThencAX = c
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573 7.3.1 First we write A = PDP
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575 7.3.19 The solution is [ e At 8
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577 7.4.12 The eigenvectors and eig
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579 9.3.30 No. They can’t be. 9.3
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581 9.8.5 We can easily see that di
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Index ∩, 529 ∪, 529 \, 529 row-
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INDEX 585 null space, 211 orthogona
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A First Course in Linear Algebra, 2017a

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