A First Course in Linear Algebra, 2017a

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320 Linear Transformations Solution. Note the matrix of this system is the same as the matrix in Example 5.66. Therefore, from Theorem 5.65, you will obtain all solutions to the above linear system by adding a particular solution ⃗x p to the solutions of the associated homogeneous system,⃗x. One particular solution is given above by ⃗x p = ⎡ ⎢ ⎣ x y z w ⎤ ⎡ ⎥ ⎦ = ⎢ ⎣ 1 1 2 1 ⎤ ⎥ ⎦ (5.6) Using this particular solution along with the solutions found in Example 5.66, we obtain the following solutions, ⎡ ⎤ ⎡ 1 3 − 4 ⎤ ⎡ ⎤ 3 1 − z 5 ⎢ 3 ⎥ ⎣ 1 ⎦ + w 2 ⎢ 3 ⎥ ⎣ 0 ⎦ + ⎢ 1 ⎥ ⎣ 2 ⎦ 0 1 1 Hence, any solution to the above linear system is of this form. ♠ Exercises Exercise 5.9.1 Write the solution set of the following system as a linear combination of vectors ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 −1 2 x 0 ⎣ 1 −2 1 ⎦⎣ y ⎦ = ⎣ 0 ⎦ 3 −4 5 z 0 Exercise 5.9.2 Using Problem 5.9.1 find the general solution to the following linear system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 −1 2 x 1 ⎣ 1 −2 1 ⎦⎣ y ⎦ = ⎣ 2 ⎦ 3 −4 5 z 4 Exercise 5.9.3 Write the solution set of the following system as a linear combination of vectors ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 −1 2 x 0 ⎣ 1 −2 1 ⎦⎣ y ⎦ = ⎣ 0 ⎦ 1 −4 5 z 0 Exercise 5.9.4 Using Problem 5.9.3 find the general solution to the following linear system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 −1 2 x 1 ⎣ 1 −2 1 ⎦⎣ y ⎦ = ⎣ −1 ⎦ 1 −4 5 z 1
5.9. The General Solution of a Linear System 321 Exercise 5.9.5 Write the solution set of the following system as a linear combination of vectors. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 −1 2 x 0 ⎣ 1 −2 0 ⎦⎣ y ⎦ = ⎣ 0 ⎦ 3 −4 4 z 0 Exercise 5.9.6 Using Problem 5.9.5 find the general solution to the following linear system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 −1 2 x 1 ⎣ 1 −2 0 ⎦⎣ y ⎦ = ⎣ 2 ⎦ 3 −4 4 z 4 Exercise 5.9.7 Write the solution set of the following system as a linear combination of vectors ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 −1 2 x 0 ⎣ 1 0 1 ⎦⎣ y ⎦ = ⎣ 0 ⎦ 1 −2 5 z 0 Exercise 5.9.8 Using Problem 5.9.7 find the general solution to the following linear system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 0 −1 2 x 1 ⎣ 1 0 1 ⎦⎣ y ⎦ = ⎣ −1 ⎦ 1 −2 5 z 1 Exercise 5.9.9 Write the solution set of the following system as a linear combination of vectors ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 0 1 1 x 0 ⎢ 1 −1 1 0 ⎥⎢ y ⎥ ⎣ 3 −1 3 2 ⎦⎣ z ⎦ = ⎢ 0 ⎥ ⎣ 0 ⎦ 3 3 0 3 w 0 Exercise 5.9.10 Using Problem 5.9.9 find the general solution to the following linear system. ⎡ ⎤⎡ ⎤ ⎡ ⎤ 1 0 1 1 x 1 ⎢ 1 −1 1 0 ⎥⎢ y ⎥ ⎣ 3 −1 3 2 ⎦⎣ z ⎦ = ⎢ 2 ⎥ ⎣ 4 ⎦ 3 3 0 3 w 3
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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
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
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Page 320 and 321: 306 Linear Transformations Example
Page 322 and 323: 308 Linear Transformations Since {T
Page 324 and 325: 310 Linear Transformations Find a b
Page 326 and 327: 312 Linear Transformations Theorem
Page 328 and 329: 314 Linear Transformations and one
Page 330 and 331: 316 Linear Transformations Hence
Page 332 and 333: 318 Linear Transformations system.
Page 338 and 339: 324 Complex Numbers Theorem 6.1: Pr
Page 340 and 341: 326 Complex Numbers Example 6.5: Co
Page 342 and 343: 328 Complex Numbers Definition 6.10
Page 344 and 345: 330 Complex Numbers Exercise 6.1.5
Page 346 and 347: 332 Complex Numbers Example 6.14: P
Page 348 and 349: 334 Complex Numbers This requires r
Page 350 and 351: 336 Complex Numbers Therefore, x 3
Page 352 and 353: 338 Complex Numbers Consider the fo
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Page 369 and 370: 7.2. Diagonalization 355 One eigenv
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Page 375 and 376: 7.2. Diagonalization 361 Theorem 7.
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Page 379 and 380: 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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