A First Course in Linear Algebra, 2017a

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104 Matrices By Lemma 2.70, this implies that the product of those E −1 j is a lower triangular matrix having all ones down the main diagonal. The above discussion and lemma gives the justification for the multiplier method. The expressions −a 21 a 11 , −a 31 a 11 ,···, −a m1 a 11 denoted respectively by M 21 ,···,M m1 to save notation which were obtained in building E 1 are the multipliers. Then according to the lemma, to find E1 −1 you simply write ⎡ ⎤ 1 0 ··· 0 −M 21 1 ··· 0 ⎢ ⎣ . . . .. ⎥ . ⎦ −M m1 0 ··· 1 Similar considerations apply to the other E −1 j . Thus L is a product of the form ⎡ ⎤ ⎤ 1 0 ··· 0 1 0 ··· 0 −M 21 1 ··· 0 0 1 ··· 0 ⎢ ⎣ . . . .. ⎥ ⎢ . ⎦···⎡ ⎣ . . 0 .. ⎥ . ⎦ −M m1 0 ··· 1 0 ··· −M m[m−1] 1 each factor having at most one nonzero column, the position of which moves from left to right in scanning the above product of matrices from left to right. It follows from what we know about the effect of multiplying on the left by an elementary matrix that the above product is of the form ⎡ ⎢ ⎣ ⎤ 1 0 ··· 0 0 −M 21 1 ··· 0 0 . −M 32 .. . . . ⎥ −M [M−1]1 . ··· 1 0 ⎦ −M M1 −M M2 ··· −M MM−1 1 In words, beginning at the left column and moving toward the right, you simply insert, into the corresponding position in the identity matrix, −1 times the multiplier which was used to zero out an entry in that position below the main diagonal in A, while retaining the main diagonal which consists entirely of ones. This is L. Exercises Exercise 2.2.1 Find an LU factorization of ⎣ Exercise 2.2.2 Find an LU factorization of ⎣ ⎡ ⎡ 1 2 0 2 1 3 1 2 3 ⎤ ⎦. 1 2 3 2 1 3 2 1 5 0 1 3 ⎤ ⎦.
Exercise 2.2.3 Find an LU factorization of the matrix ⎣ Exercise 2.2.4 Find an LU factorization of the matrix ⎣ Exercise 2.2.5 Find an LU factorization of the matrix ⎣ Exercise 2.2.6 Find an LU factorization of the matrix ⎣ Exercise 2.2.7 Find an LU factorization of the matrix Exercise 2.2.8 Find an LU factorization of the matrix Exercise 2.2.9 Find an LU factorization of the matrix ⎡ ⎡ ⎡ ⎡ ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ 1 −2 −5 0 −2 5 11 3 3 −6 −15 1 1 −1 −3 −1 −1 2 4 3 2 −3 −7 −3 1 −3 −4 −3 −3 10 10 10 1 −6 2 −5 1 3 1 −1 3 10 8 −1 2 5 −3 −3 3 −2 1 9 −8 6 −6 2 2 3 2 −7 −3 −1 3 9 9 −12 3 19 −16 12 40 −26 −1 −3 −1 1 3 0 3 9 0 4 12 16 ⎤ ⎥ ⎦ . ⎤ ⎦. 2.2. LU Factorization 105 Exercise 2.2.10 Find the LU factorization of the coefficient matrix using Dolittle’s method and use it to solve the system of equations. x + 2y = 5 2x + 3y = 6 ⎤ ⎤ ⎥ ⎦ . ⎥ ⎦ . ⎤ ⎦. ⎤ ⎦. ⎤ ⎦. Exercise 2.2.11 Find the LU factorization of the coefficient matrix using Dolittle’s method and use it to solve the system of equations. x + 2y + z = 1 y + 3z = 2 2x + 3y = 6
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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 68 and 69: 54 Matrices Consider the following
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
Page 92 and 93: 78 Matrices What if the right side,
Page 94 and 95: 80 Matrices First, consider the fol
Page 96 and 97: 82 Matrices You can see that B is t
Page 98 and 99: 84 Matrices = = ⎡ ⎣ ⎡ ⎣ 1 0
Page 100 and 101: 86 Matrices First: ⎡ ⎣ 0 1 0 1
Page 102 and 103: 88 Matrices Proof. Suppose that A a
Page 104 and 105: 90 Matrices It follows that the red
Page 106 and 107: 92 Matrices (h) DE ⎡ Exercise 2.1
Page 108 and 109: 94 Matrices (a) (A − B) 2 = A 2
Page 110 and 111: 96 Matrices Exercise 2.1.42 Let ⎡
Page 112 and 113: 98 Matrices (a) Find the elementary
Page 114 and 115: 100 Matrices One way to find the LU
Page 116 and 117: 102 Matrices which yields very quic
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
Page 126 and 127: 112 Determinants The following prov
Page 128 and 129: 114 Determinants 3.1.3 Properties o
Page 130 and 131: 116 Determinants Example 3.23: Mult
Page 132 and 133: 118 Determinants Solution. Consider
Page 134 and 135: 120 Determinants = ∑a 1,l (cof(B)
Page 136 and 137: 122 Determinants Theorem 3.35: Let
Page 138 and 139: 124 Determinants Here, det(C)=−3d
Page 140 and 141: 126 Determinants (a) minor(A) 11 (b
Page 142 and 143: 128 Determinants Exercise 3.1.13 An
Page 144 and 145: 130 Determinants 3.2.1 A Formula fo
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Page 148 and 149: 134 Determinants The cofactor matri
Page 150 and 151: 136 Determinants Therefore, ∣ ∣
Page 152 and 153: 138 Determinants After row operatio
Page 154 and 155: 140 Determinants Using the given po
Page 156 and 157: 142 Determinants Does there exist a
Page 158 and 159: 144 R n Hence, every element in R 2
Page 160 and 161: 146 R n components are equal. Preci
Page 162 and 163: 148 R n Theorem 4.6: Properties of
Page 164 and 165: 150 R n 4.3 Geometric Meaning of Ve
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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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A First Course in Linear Algebra, 2017a

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