A First Course in Linear Algebra, 2017a

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78 Matrices What if the right side, B, of2.10 had been ⎣ ⎡ ⎣ ⎡ 0 1 3 1 0 1 1 −1 1 1 1 −1 ⎤ ⎦? In other words, what would be the solution to ⎤⎡ ⎦⎣ x y z ⎤ ⎡ ⎦ = ⎣ 0 1 3 ⎤ ⎦? By the above discussion, the solution is given by ⎡ ⎡ ⎤ 1 1 0 x 2 2 ⎣ y ⎦ = A −1 B = ⎢ ⎣ 1 −1 0 z 1 − 1 2 − 1 2 ⎤ ⎡ ⎥⎣ ⎦ 0 1 3 ⎤ ⎡ ⎦ = ⎣ 2 −1 −2 ⎤ ⎦ This illustrates that for a system AX = B where A −1 exists, it is easy to find the solution when the vector B is changed. We conclude this section with some important properties of the inverse. Theorem 2.41: Inverses of Transposes and Products Let A,B,andA i for i = 1,...,k be n × n matrices. 1. If A is an invertible matrix, then (A T ) −1 =(A −1 ) T 2. If A and B are invertible matrices, then AB is invertible and (AB) −1 = B −1 A −1 3. If A 1 ,A 2 ,...,A k are invertible, then the product A 1 A 2 ···A k is invertible, and (A 1 A 2 ···A k ) −1 = A −1 k A−1 k−1 ···A−1 2 A−1 1 Consider the following theorem. Theorem 2.42: Properties of the Inverse Let A be an n × n matrix and I the usual identity matrix. 1. I is invertible and I −1 = I 2. If A is invertible then so is A −1 ,and(A −1 ) −1 = A 3. If A is invertible then so is A k ,and(A k ) −1 =(A −1 ) k 4. If A is invertible and p is a nonzero real number, then pA is invertible and (pA) −1 = 1 p A−1
2.1. Matrix Arithmetic 79 2.1.9 Elementary Matrices We now turn our attention to a special type of matrix called an elementary matrix.Anelementarymatrix is always a square matrix. Recall the row operations given in Definition 1.11. Any elementary matrix, which we often denote by E, is obtained from applying one row operation to the identity matrix of the same size. For example, the matrix E = [ 0 1 1 0 is the elementary matrix obtained from switching the two rows. The matrix ⎡ 1 0 ⎤ 0 E = ⎣ 0 3 0 ⎦ 0 0 1 is the elementary matrix obtained from multiplying the second row of the 3 × 3 identity matrix by 3. The matrix [ ] 1 0 E = −3 1 is the elementary matrix obtained from adding −3 times the first row to the third row. You may construct an elementary matrix from any row operation, but remember that you can only apply one operation. Consider the following definition. Definition 2.43: Elementary Matrices and Row Operations Let E be an n × n matrix. Then E is an elementary matrix if it is the result of applying one row operation to the n × n identity matrix I n . Those which involve switching rows of the identity matrix are called permutation matrices. ] Therefore, E constructed above by switching the two rows of I 2 is called a permutation matrix. Elementary matrices can be used in place of row operations and therefore are very useful. It turns out that multiplying (on the left hand side) by an elementary matrix E will have the same effect as doing the row operation used to obtain E. The following theorem is an important result which we will use throughout this text. Theorem 2.44: Multiplication by an Elementary Matrix and Row Operations To perform any of the three row operations on a matrix A it suffices to take the product EA,where E is the elementary matrix obtained by using the desired row operation on the identity matrix. Therefore, instead of performing row operations on a matrix A, we can row reduce through matrix multiplication with the appropriate elementary matrix. We will examine this theorem in detail for each of the three row operations given in Definition 1.11.
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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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Page 70 and 71: 56 Matrices Solution. Notice that b
Page 72 and 73: 58 Matrices Proposition 2.10: Prope
Page 74 and 75: 60 Matrices on the left by a vector
Page 76 and 77: 62 Matrices will satisfy the equati
Page 78 and 79: 64 Matrices Solution. First check i
Page 80 and 81: 66 Matrices Solution. First check i
Page 82 and 83: 68 Matrices Proposition 2.25: Prope
Page 84 and 85: 70 Matrices Definition 2.29: Symmet
Page 86 and 87: 72 Matrices Theorem 2.34: Uniquenes
Page 88 and 89: 74 Matrices Let’s solve the first
Page 90 and 91: 76 Matrices Notice that the left ha
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Page 96 and 97: 82 Matrices You can see that B is t
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Page 100 and 101: 86 Matrices First: ⎡ ⎣ 0 1 0 1
Page 102 and 103: 88 Matrices Proof. Suppose that A a
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Page 106 and 107: 92 Matrices (h) DE ⎡ Exercise 2.1
Page 108 and 109: 94 Matrices (a) (A − B) 2 = A 2
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Page 120 and 121: 106 Matrices Exercise 2.2.12 Find t
Page 122 and 123: 108 Determinants The 2×2 determina
Page 124 and 125: 110 Determinants Definition 3.7: Th
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Page 128 and 129: 114 Determinants 3.1.3 Properties o
Page 130 and 131: 116 Determinants Example 3.23: Mult
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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
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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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