A First Course in Linear Algebra, 2017a

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128 Determinants Exercise 3.1.13 An operation is done to get from the first matrix to the second. Identify what was done and tell how it will affect the value of the determinant. [ ] [ ] a b b a →···→ c d d c Exercise 3.1.14 Let A be an r × r matrix and suppose there are r − 1 rows (columns) such that all rows (columns) are linear combinations of these r − 1 rows (columns). Show det(A)=0. Exercise 3.1.15 Show det(aA)=a n det(A) for an n × n matrix A and scalar a. Exercise 3.1.16 Construct 2 × 2 matrices A and B to show that the detAdetB = det(AB). Exercise 3.1.17 Is it true that det(A + B)=det(A)+det(B)? If this is so, explain why. If it is not so, give a counter example. Exercise 3.1.18 An n × n matrix is called nilpotent if for some positive integer, k it follows A k = 0. If A is a nilpotent matrix and k is the smallest possible integer such that A k = 0, what are the possible values of det(A)? Exercise 3.1.19 A matrix is said to be orthogonal if A T A = I. Thus the inverse of an orthogonal matrix is just its transpose. What are the possible values of det(A) if A is an orthogonal matrix? Exercise 3.1.20 Let A and B be two n × n matrices. A ∼ B(Aissimilar to B) means there exists an invertible matrix P such that A = P −1 BP. Show that if A ∼ B, then det(A)=det(B). Exercise 3.1.21 Tell whether each statement is true or false. If true, provide a proof. If false, provide a counter example. (a) If A is a 3 × 3 matrix with a zero determinant, then one column must be a multiple of some other column. (b) If any two columns of a square matrix are equal, then the determinant of the matrix equals zero. (c) For two n × n matrices A and B, det(A + B)=det(A)+det(B). (d) For an n × nmatrixA,det(3A)=3det(A) (e) If A −1 exists then det ( A −1) = det(A) −1 . (f) If B is obtained by multiplying a single row of A by 4 then det(B)=4det(A). (g) For A an n × nmatrix,det(−A)=(−1) n det(A). (h) If A is a real n × n matrix, then det ( A T A ) ≥ 0.
3.2. Applications of the Determinant 129 (i) If A k = 0 for some positive integer k, then det(A)=0. (j) If AX = 0 for some X ≠ 0, then det(A)=0. Exercise 3.1.22 Find the determinant using row operations to first simplify. 1 2 1 2 3 2 ∣ −4 1 2 ∣ Exercise 3.1.23 Find the determinant using row operations to first simplify. 2 1 3 2 4 2 ∣ 1 4 −5 ∣ Exercise 3.1.24 Find the determinant using row operations to first simplify. 1 2 1 2 3 1 −2 3 −1 0 3 1 ∣ 2 3 2 −2 ∣ Exercise 3.1.25 Find the determinant using row operations to first simplify. 1 4 1 2 3 2 −2 3 −1 0 3 3 ∣ 2 1 2 −2 ∣ 3.2 Applications of the Determinant Outcomes A. Use determinants to determine whether a matrix has an inverse, and evaluate the inverse using cofactors. B. Apply Cramer’s Rule to solve a 2 × 2 or a 3 × 3 linear system. C. Given data points, find an appropriate interpolating polynomial and use it to estimate points.
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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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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 116 and 117: 102 Matrices which yields very quic
Page 118 and 119: 104 Matrices By Lemma 2.70, this im
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
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Page 136 and 137: 122 Determinants Theorem 3.35: Let
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Page 140 and 141: 126 Determinants (a) minor(A) 11 (b
Page 144 and 145: 130 Determinants 3.2.1 A Formula fo
Page 146 and 147: 132 Determinants You can verify tha
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
Page 166 and 167: 152 R n 4.4 Length of a Vector Outc
Page 168 and 169: 154 R n Solution. We will use the f
Page 170 and 171: 156 R n = √ 1 + 9 + 16 = √ 26 I
Page 172 and 173: 158 R n ⃗u 2⃗v ⃗u + 2⃗v ⃗
Page 174 and 175: 160 R n ⎡ Let ⃗q = ⎣ Then, x
Page 176 and 177: 162 R n Let t = x−2 y−1 3 ,t =
Page 178 and 179: 164 R n Example 4.26: Compute a Dot
Page 180 and 181: 166 R n Notice that this proof was
Page 182 and 183: 168 R n Therefore, the cosine of th
Page 184 and 185: 170 R n 4.7.3 Projections In some a
Page 186 and 187: 172 R n We will conclude this secti
Page 188 and 189: 174 R n Exercise 4.7.11 Find proj
Page 190 and 191: 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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