A First Course in Linear Algebra, 2017a

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584 INDEX field axioms, 323 finite dimensional, 483 force, 257 free variable, 25 Fundamental Theorem of Algebra, 323 Gauss-Jordan Elimination, 24 Gaussian Elimination, 24 general solution, 319 solution space, 318 hyper-planes, 6 idempotent, 93 identity transformation, 267, 494 image, 512 improper subspace, 480 included angle, 167 inconsistent system, 8 induction hypothesis, 532 injection, 288 inner product, 163 intersection, 492, 529 intervals notation, 530 invertible matrices isomorphism, 505 isomorphic, 293, 502 equivalence relation, 296 isomorphism, 293, 502 bases, 296 composition, 296 equivalence, 297, 506 inverse, 295 invertible matrices, 505 kernel, 211, 317, 512 Kirchhoff’s law, 38 Kronecker symbol, 71, 235 Laplace expansion, 110 leading entry, 16 least square approximation, 247 linear combination, 30, 148 linear dependence, 190 linear independence, 191 enlarging to form a basis, 486 linear independent, 229 linear map, 293 defining on a basis, 299 image, 305 kernel, 305 linear transformation, 266, 293, 493 composite, 279 image, 287 matrix, 269 range, 287 linearly dependent, 469 linearly independent, 469 lines parametric equation, 161 symmetric form, 161 vector equation, 159 LU decomposition non existence, 99 LU factorization, 99 by inspection, 100 justification, 102 solving systems, 101 main diagonal, 112 Markov matrices, 371 mathematical induction, 531, 532 matrix, 14, 53 addition, 55 augmented matrix, 14, 15 coefficient matrix, 14 column space, 207 commutative, 67 components of a matrix, 54 conformable, 62 diagonal matrix, 357 dimension, 14 entries of a matrix, 54 equality, 54 equivalent, 27 finding the inverse, 74 identity, 71 improper, 236 inverse, 71 invertible, 71 kernel, 211 lower triangular, 112 main diagonal, 357
INDEX 585 null space, 211 orthogonal, 128, 234 orthogonally diagonalizable, 368 proper, 236 properties of addition, 56 properties of scalar multiplication, 58 properties of transpose, 69 raising to a power, 367 rank, 31 row space, 207 scalar multiplication, 57 skew symmetric, 70 square, 54 symmetric, 70 transpose, 69 upper triangular, 112 matrix exponential, 385 matrix form AX=B, 61 matrix multiplication, 62 ijth entry, 65 properties, 68 vectors, 60 migration matrix, 371 multiplicity, 344 multipliers, 104 Newton, 257 nilpotent, 128 non defective, 401 nontrivial solution, 28 null space, 211, 317 nullity, 215, 307 one to one, 288 linear independence, 296 onto, 288 orthogonal, 231 orthogonal complement, 242 orthogonality and minimization, 247 orthogonally diagonalizable, 401 orthonormal, 231 parallelepiped, 184 volume, 184 parameter, 23 particular solution, 317 permutation matrices, 79 pivot column, 18 pivot position, 18 plane normal vector, 175 scalar equation, 176 vector equation, 175 polar coordinates, 433 polynomials factoring, 335 position vector, 144 positive definite, 411 Cholesky factorization, 414 invertible, 411 principal submatrices, 413 principal axes, 398, 423 principal axis quadratic forms, 421 principal submatrices, 412 positive definite, 413 proper subspace, 197, 480 QR factorization, 416 quadratic form, 421 quadratic formula, 337 random walk, 374 range of matrix transformation, 287 rank, 307 rank added to nullity, 307 reflection across a given vector, 287 regression line, 250 resultant, 258 right handed system, 178 row operations, 15, 114 row space, 207 scalar, 8 scalar product, 163 scalar transformation, 494 scalars, 147 scaling factor, 419 set notation, 529 similar matrices, 355 similar matrix, 350 singular values, 403 skew lines, 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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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
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A.2. Well Ordering and Induction 53
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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 and 596: 581 9.8.5 We can easily see that di
Page 597: Index ∩, 529 ∪, 529 \, 529 row-
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