A First Course in Linear Algebra, 2017a

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582 Selected Exercise Answers 9.9.7 ⎡ ⎣ 5 3 2 2 3 5 5 5 −2 ⎤⎡ ⎦⎣ 11 4 1 10 4 1 12 3 1 ⎤ ⎡ ⎦ = ⎣ 109 38 10 112 35 10 81 34 8 9.9.11 Recall that proj ⃗u (⃗v)= ⃗v•⃗u ⃗u andsothedesiredmatrixhasi th column equal to proj ‖⃗u‖ 2 ⃗u (⃗e i ). Therefore, the matrix desired is ⎡ ⎤ 1 −2 3 1 ⎣ −2 4 −6 ⎦ 14 3 −6 9 ⎤ ⎦ 9.9.12 9.9.13 9.9.15 C B (⃗x)= ⎣ 9.9.16 M B2 B 1 = ⎡ [ 2 1 −1 ⎤ ⎦. ] 1 0 −1 1 ⎡ 1 ⎣ 35 ⎡ 1 ⎣ 10 1 5 3 5 25 15 3 15 9 1 0 3 0 0 0 3 0 9 ⎤ ⎦ ⎤ ⎦
Index ∩, 529 ∪, 529 \, 529 row-echelon form, 16 reduced row-echelon form, 16 algorithm, 19 adjugate, 130 back substitution, 13 base case, 532 basic solution, 29 basic variable, 25 basis, 199, 200, 230, 481 any two same size, 483 box product, 184 cardioid, 437 Cauchy Schwarz inequality, 165 characteristic equation, 343 chemical reactions balancing, 33 Cholesky factorization positive definite, 414 classical adjoint, 130 Cofactor Expansion, 110 cofactor matrix, 130 column space, 207 complex eigenvalues, 362 complex numbers absolute value, 328 addition, 323 argument, 330 conjugate, 325 conjugate of a product, 330 modulus, 328, 330 multiplication, 324 polar form, 330 roots, 333 standard form, 323 triangle inequality, 328 component form, 159 component of a force, 261, 263 consistent system, 8 coordinate isomorphism, 517 coordinate vector, 311, 518 Cramer’s rule, 135 cross product, 178, 180 area of parallelogram, 182 coordinate description, 180 geometric description, 179, 180 cylindrical coordinates, 443 De Moivre’s theorem, 333 determinant, 107 cofactor, 109 expanding along row or column, 110 matrix inverse formula, 130 minor, 108 product, 116, 121 row operations, 114 diagonalizable, 357, 401 dimension, 201 dimension of vector space, 483 direct sum, 492 direction vector, 159 distance formula, 153 properties, 154 dot product, 163 properties, 164 eigenvalue, 342 eigenvalues calculating, 344 eigenvector, 342 eigenvectors calculating, 344 elementary matrix, 79 inverse, 82 elementary operations, 9 elementary row operations, 15 empty set, 529 equivalence relation, 355 exchange theorem, 200, 481 extending a basis, 205 583
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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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202 R n ⃗u 1 = Furthermore, ⎡
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204 R n ⎡ ⎢ ⎣ 1 2 1 1 3 0 1 3
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206 R n Solution. An easy way to do
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208 R n Since {⃗r 1 ,..., p⃗r j
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210 R n Example 4.101: Rank, Column
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212 R n Definition 4.106: Image of
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214 R n Example 4.110: Null Space o
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216 R n Theorem 4.114: Let A be an
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218 R n Exercise 4.10.7 Here are so
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220 R n Exercise 4.10.17 Are the fo
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222 R n Thse vectors can’t possib
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224 R n ⎧⎡ ⎪⎨ Exercise 4.10
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226 R n Exercise 4.10.52 Consider t
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228 R n Exercise 4.10.68 Find the r
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230 R n Solution. We already verifi
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232 R n Thus to find a correspondin
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234 R n That is: We readily compute
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236 R n We will say that the column
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238 R n To show that {⃗v 1 ,··
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240 R n is an orthogonal basis of U
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242 R n = = = ⎡ ⎣ ⎡ ⎣ ⎡
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244 R n In order to find W ⊥ , we
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246 R n Now, we need to write ⃗y
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248 R n Theorem 4.150: Existence of
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250 R n Solution. First, consider w
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252 R n −1 1 2 3 4 5 6 y 4 2 x Re
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254 R n Exercise 4.11.7 For U an or
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256 R n and here is a point: (1,1,2
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258 R n What does addition of vecto
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260 R n 4 3 First we want to know t
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262 R n = −58 Newton meters Note
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264 R n Exercise 4.12.19 A girl dra
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266 Linear Transformations numerica
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268 Linear Transformations Exercise
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270 Linear Transformations In this
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272 Linear Transformations ⎡ ⎣
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274 Linear Transformations ⎡ T
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276 Linear Transformations ⎡ (b)
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278 Linear Transformations The nece
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280 Linear Transformations Then, co
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282 Linear Transformations More gen
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284 Linear Transformations Solution
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288 Linear Transformations Definiti
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290 Linear Transformations This tel
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292 Linear Transformations Example
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294 Linear Transformations 2. T is
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296 Linear Transformations Proposit
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298 Linear Transformations Proof. S
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300 Linear Transformations You can
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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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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
Page 545 and 546: A.2. Well Ordering and Induction 53
Page 547 and 548: A.2. Well Ordering and Induction 53
Page 549 and 550: Appendix B Selected Exercise Answer
Page 551 and 552: 537 1.2.33 Solution is: [x = 2 −
Page 553 and 554: 539 2.1.7 0A =(0 + 0)A = 0A + 0A.No
Page 555 and 556: 541 [ 2.1.18 Let A = 2.1.20 Let A =
Page 557 and 558: 543 2.1.40 2.1.41 ⎡ ⎣ ⎡ ⎣ 1
Page 559 and 560: 545 2.2.10 An LU factorization of t
Page 561 and 562: 547 3.1.14 If the determinant is no
Page 563 and 564: 549 ⎡ 3.2.2 det⎣ 1 2 0 0 2 1 3
Page 565 and 566: 551 3.2.16 This follows because det
Page 567 and 568: 553 4.7.16 ( ) ( ) ⃗u •⃗v ⃗
Page 569 and 570: 555 It follows that the expression
Page 571 and 572: 557 4.11.6 (a) Orthogonal. (b) Symm
Page 573 and 574: 559 4.11.14 Then a solution is ⎡
Page 575 and 576: 561 4.12.23 ⎡ ⎢ ⃗F 1 • ⎣
Page 577 and 578: 563 5.2.13 ⎡ 1 ⎣ 10 1 0 3 0 0 0
Page 579 and 580: 565 Now to write in terms of (a,b),
Page 581 and 582: 567 ⎡ 5.9.7 Solution is: ⎣ −
Page 583 and 584: 569 6.1.6 If p(z)=0, then you have
Page 585 and 586: 571 7.1.2 Say AX = λX.ThencAX = c
Page 587 and 588: 573 7.3.1 First we write A = PDP
Page 589 and 590: 575 7.3.19 The solution is [ e At 8
Page 591 and 592: 577 7.4.12 The eigenvectors and eig
Page 593 and 594: 579 9.3.30 No. They can’t be. 9.3
Page 595: 581 9.8.5 We can easily see that di
Page 599 and 600: INDEX 585 null space, 211 orthogona
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A First Course in Linear Algebra, 2017a

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