A First Course in Linear Algebra, 2017a

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Table of Contents Table of Contents iii Preface 1 1 Systems of Equations 3 1.1 Systems of Equations, Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Systems of Equations, Algebraic Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 Uniqueness of the Reduced Row-Echelon Form . . . . . . . . . . . . . . . . . . 25 1.2.4 Rank and Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.5 Balancing Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.2.6 Dimensionless Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.7 An Application to Resistor Networks . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Matrices 53 2.1 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.1 Addition of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.2 Scalar Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.1.3 Multiplication of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.1.4 The ij th Entry of a Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.1.5 Properties of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1.6 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.7 The Identity and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.8 Finding the Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.1.9 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.10 More on Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.2.1 Finding An LU Factorization By Inspection . . . . . . . . . . . . . . . . . . . . . 99 2.2.2 LU Factorization, Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 100 2.2.3 Solving Systems using LU Factorization . . . . . . . . . . . . . . . . . . . . . . . 101 2.2.4 Justification for the Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . 102 iii
Page 1: with Open Texts Linear A Algebra wi
Page 5: A First Course in Linear Algebra an
Page 9: A First Course in Linear Algebra an
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
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Page 50 and 51: 36 Systems of Equations Here m deno
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Page 54 and 55: 40 Systems of Equations resistors g
Page 56 and 57: 42 Systems of Equations Exercises E
Page 58 and 59: 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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