A First Course in Linear Algebra, 2017a

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A First Course in Linear Algebra an Open Text Base Text Revision History: Version 2021 — Revision A Extensive edits, additions, and revisions have been completed by the editorial staff at Lyryx Learning. All new content (text and images) is released under the same license as noted above. 2021 A • Lyryx: Front matter has been updated including cover, Lyryx with Open Texts, copyright, and revision pages. Attribution page has been added. • Lyryx: Typo and other minor fixes have been implemented throughout. 2017 A • Lyryx: Front matter has been updated including cover, copyright, and revision pages. • I. Farah: contributed edits and revisions, particularly the proofs in the Properties of Determinants II: Some Important Proofs section 2016 B • Lyryx: The text has been updated with the addition of subsections on Resistor Networks and the Matrix Exponential based on original material by K. Kuttler. • Lyryx: New example on Random Walks developed. 2016 A • Lyryx: The layout and appearance of the text has been updated, including the title page and newly designed back cover. 2015 A • Lyryx: The content was modified and adapted with the addition of new material and several images throughout. • Lyryx: Additional examples and proofs were added to existing material throughout. 2012 A • Original text by K. Kuttler of Brigham Young University. That version is used under Creative Commons license CC BY (https://creativecommons.org/licenses/by/3.0/) made possible by funding from The Saylor Foundation’s Open Textbook Challenge. See Elementary Linear Algebra for more information and the original version.
Page 1: with Open Texts Linear A Algebra wi
Page 5: A First Course in Linear Algebra an
Page 11 and 12: Table of Contents Table of Contents
Page 13 and 14: v 4.11.3 Gram-Schmidt Process . . .
Page 15: Preface A First Course in Linear Al
Page 18 and 19: 4 Systems of Equations Example 1.1:
Page 20 and 21: 6 Systems of Equations planes so th
Page 22 and 23: 8 Systems of Equations Definition 1
Page 24 and 25: 10 Systems of Equations Consider th
Page 26 and 27: 12 Systems of Equations 2. Next we
Page 28 and 29: 14 Systems of Equations 1.2.2 Gauss
Page 30 and 31: 16 Systems of Equations Note how th
Page 32 and 33: 18 Systems of Equations Definition
Page 34 and 35: 20 Systems of Equations 2. Step two
Page 36 and 37: 22 Systems of Equations There is no
Page 38 and 39: 24 Systems of Equations Take −1 t
Page 40 and 41: 26 Systems of Equations Example 1.2
Page 42 and 43: 28 Systems of Equations all three h
Page 44 and 45: 30 Systems of Equations Example 1.3
Page 46 and 47: 32 Systems of Equations rank of the
Page 48 and 49: 34 Systems of Equations which we ca
Page 50 and 51: 36 Systems of Equations Here m deno
Page 52 and 53: 38 Systems of Equations 1.2.7 An Ap
Page 54 and 55: 40 Systems of Equations resistors g
Page 56 and 57: 42 Systems of Equations Exercises E
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44 Systems of Equations [ 1 2 2 2 h
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46 Systems of Equations Exercise 1.
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48 Systems of Equations Exercise 1.
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50 Systems of Equations 3Ω 20 volt
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Chapter 2 Matrices 2.1 Matrix Arith
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2.1. Matrix Arithmetic 55 In other
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2.1. Matrix Arithmetic 57 2.1.2 Sca
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2.1. Matrix Arithmetic 59 In this c
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2.1. Matrix Arithmetic 61 follows.
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2.1. Matrix Arithmetic 63 Definitio
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2.1. Matrix Arithmetic 65 The j th
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2.1. Matrix Arithmetic 67 Substitut
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2.1. Matrix Arithmetic 69 Definitio
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2.1. Matrix Arithmetic 71 2.1.7 The
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2.1. Matrix Arithmetic 73 [ ]) = A
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2.1. Matrix Arithmetic 75 This algo
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2.1. Matrix Arithmetic 77 AX = B. S
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2.1. Matrix Arithmetic 79 2.1.9 Ele
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2.1. Matrix Arithmetic 81 Example 2
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2.1. Matrix Arithmetic 83 This equa
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2.1. Matrix Arithmetic 85 → ⎡
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2.1. Matrix Arithmetic 87 2.1.10 Mo
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2.1. Matrix Arithmetic 89 Example 2
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2.1. Matrix Arithmetic 91 Exercise
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2.1. Matrix Arithmetic 93 ⎡ in th
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2.2. LU Factorization 99 2.2 LU Fac
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2.2. LU Factorization 101 Thenextst
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2.2. LU Factorization 103 For a sim
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Exercise 2.2.3 Find an LU factoriza
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Chapter 3 Determinants 3.1 Basic Te
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3.1. Basic Techniques and Propertie
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3.2. Applications of the Determinan
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Chapter 4 R n 4.1 Vectors in R n Ou
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4.1. Vectors in R n 145 Now, imagin
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4.2. Algebra in R n 147 To add vect
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4.2. Algebra in R n 149 is a linear
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4.3. Geometric Meaning of Vector Ad
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4.4. Length of a Vector 153 P =(p 1
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4.4. Length of a Vector 155 We can
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4.5. Geometric Meaning of Scalar Mu
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4.6. Parametric Lines 159 This equa
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4.6. Parametric Lines 161 This set
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4.7. The Dot Product 163 Exercise 4
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4.7. The Dot Product 165 Solution.
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4.7. The Dot Product 167 4.7.2 The
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4.7. The Dot Product 169 Example 4.
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4.7. The Dot Product 171 Definition
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4.7. The Dot Product 173 Exercises
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4.8. Planes in R n 175 4.8 Planes i
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4.8. Planes in R n 177 Consider the
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4.9. The Cross Product 179 Consider
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4.9. The Cross Product 181 There is
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4.9. The Cross Product 183 Solution
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4.9. The Cross Product 185 Example
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4.9. The Cross Product 187 Exercise
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4.10. Spanning, Linear Independence
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Exercise 4.10.46 Let M = ⎧ ⎪⎨
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4.11. Orthogonality and the Gram Sc
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4.12. Applications 257 4.12 Applica
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4.12. Applications 259 of ⃗u corr
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4.12. Applications 261 For any forc
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4.12. Applications 263 Exercise 4.1
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Chapter 5 Linear Transformations 5.
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5.1. Linear Transformations 267 Sol
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5.2. The Matrix of a Linear Transfo
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5.3. Properties of Linear Transform
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5.3. Properties of Linear Transform
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Exercises 5.4. Special Linear Trans
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5.4. Special Linear Transformations
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5.4. Special Linear Transformations
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5.5. One to One and Onto Transforma
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5.6. Isomorphisms 293 Exercise 5.5.
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5.6. Isomorphisms 295 3. T is onto:
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5.6. Isomorphisms 297 Proof. First
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5.6. Isomorphisms 299 Now suppose t
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5.6. Isomorphisms 301 Hence, A =
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5.6. Isomorphisms 303 Exercise 5.6.
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5.7 The Kernel And Image Of A Linea
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5.7. The Kernel And Image Of A Line
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5.7. The Kernel And Image Of A Line
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5.9. The General Solution of a Line
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⃗x = ⎡ ⎢ ⎣ x y z w ⎤ ⎥
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5.9. The General Solution of a Line
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Chapter 6 Complex Numbers 6.1 Compl
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6.1. Complex Numbers 325 Theorem 6.
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6.1. Complex Numbers 327 Example 6.
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6.1. Complex Numbers 329 Hence, bot
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6.2. Polar Form 331 Definition 6.12
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6.3. Roots of Complex Numbers 333 6
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6.3. Roots of Complex Numbers 335 E
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6.4. The Quadratic Formula 337 6.4
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6.4. The Quadratic Formula 339 (b)
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342 Spectral Theory In this case, t
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344 Spectral Theory 7.1.2 Finding E
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346 Spectral Theory The solution is
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348 Spectral Theory It is a good id
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350 Spectral Theory This clearly eq
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352 Spectral Theory Through using e
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354 Spectral Theory ⎡ Find A ⎣
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356 Spectral Theory 7.2.1 Similarit
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358 Spectral Theory Notice that the
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360 Spectral Theory Next, we need t
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362 Spectral Theory In this case, t
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364 Spectral Theory Therefore, the
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366 Spectral Theory Exercise 7.2.9
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368 Spectral Theory Then also which
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370 Spectral Theory The reduced row
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372 Spectral Theory ♠ 7.3.3 Marko
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374 Spectral Theory = ⎡ ⎣ 102 1
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376 Spectral Theory Example 7.37: P
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378 Spectral Theory Eigenvalues of
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380 Spectral Theory Example 7.42: S
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382 Spectral Theory and that the ei
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384 Spectral Theory Note that the e
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386 Spectral Theory 7.3.5 The Matri
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388 Spectral Theory The matrix expo
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390 Spectral Theory and Lastly, AX
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392 Spectral Theory It is unknown w
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394 Spectral Theory the top row bei
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396 Spectral Theory Theorem 7.50: O
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398 Spectral Theory showing D is re
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400 Spectral Theory Solution. You c
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402 Spectral Theory However, it is
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404 Spectral Theory Since λ ≠ 0a
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406 Spectral Theory where σ is giv
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408 Spectral Theory and Therefore,
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410 Spectral Theory ⎡ A = ⎣ −
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412 Spectral Theory where λ 1 ,λ
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414 Spectral Theory The Cholesky Fa
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416 Spectral Theory 7.4.4 QR Factor
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418 Spectral Theory Finally, constr
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420 Spectral Theory Example 7.89: F
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422 Spectral Theory = a 11 x 2 1 +
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424 Spectral Theory = (U⃗y) T A(U
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426 Spectral Theory Example 7.95: C
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430 Spectral Theory Supply reasons
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434 Some Curvilinear Coordinate Sys
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450 Vector Spaces Definition 9.2: A
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452 Vector Spaces ⎡ ⎤ ⎡ ⎤ x
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454 Vector Spaces ⎡ = a⎢ ⎣ =
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456 Vector Spaces We now consider s
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458 Vector Spaces • Finally, we s
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460 Vector Spaces B + A = B [ ] 0 0
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462 Vector Spaces 1. When we say th
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464 Vector Spaces Is this a special
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466 Vector Spaces Definition 9.12:
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468 Vector Spaces For this to be tr
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470 Vector Spaces The augmented mat
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472 Vector Spaces Proof. Suppose
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474 Vector Spaces If the above thre
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476 Vector Spaces If these are line
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480 Vector Spaces { } 2. Let ⃗w 1
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482 Vector Spaces Say c k ≠ 0. Th
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484 Vector Spaces Example 9.38: Dim
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486 Vector Spaces Let B = {[ 1 0 0
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488 Vector Spaces and not all of th
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490 Vector Spaces Now the following
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492 Vector Spaces 9.5 Sums and Inte
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494 Vector Spaces Example 9.56: Lin
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496 Vector Spaces Therefore, = 1 2
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498 Vector Spaces ⎡ T ⎣ 0 −1
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500 Vector Spaces Consider the foll
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502 Vector Spaces Example 9.68: Ont
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504 Vector Spaces This clearly only
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506 Vector Spaces Since T is one to
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508 Vector Spaces there would exist
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510 Vector Spaces Exercise 9.7.7 De
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512 Vector Spaces 9.8 The Kernel An
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514 Vector Spaces The values of a,b
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516 Vector Spaces dimension of im(T
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520 Vector Spaces We now discuss th
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522 Vector Spaces Therefore C B2 [T
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524 Vector Spaces Consider the foll
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526 Vector Spaces Exercise 9.9.3 Su
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528 Vector Spaces ⎡ (a) T ⎣ ⎡
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530 Some Prerequisite Topics • [a
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532 Some Prerequisite Topics This p
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534 Some Prerequisite Topics Suppos
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536 Selected Exercise Answers 1.2.1
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540 Selected Exercise Answers (d) (
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542 Selected Exercise Answers (c) [
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546 Selected Exercise Answers ⎡ S
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556 Selected Exercise Answers 4.10.
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558 Selected Exercise Answers This
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560 Selected Exercise Answers 4.12.
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562 Selected Exercise Answers = [
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566 Selected Exercise Answers {[ ]}
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574 Selected Exercise Answers Letti
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578 Selected Exercise Answers Then
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584 INDEX field axioms, 323 finite
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586 INDEX solution space, 317 span,
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A First Course in Linear Algebra, 2017a

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