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MAT2611linear_al_done_right_notes.
, Solutions for exercises/Notes for Linear
Algebra Done Right by Sheldon Axler
Toan Quang Pham
mathtangents@gmail.com
th
Monday 10 September, 2018
Contents
1. Some note before reading the book 4
2. Terminology 4
3. Chapter 1 - Vector Spaces 4
3.1. Excercises 1.B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2. 1.C Subspaces.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3. Exercises 1.C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. Chapter 2 - Finite-dimensional vector spaces 9
4.1. 2.A Span and linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1.1. Main theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.1.2. Important/Interesting results from Exercise 2.A . . .. . . . . . . . . . . . 11
4.2. Exercises 2.A .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3. 2.B Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4. Addition, scalar multiplication of specific list of vectors . . . .. . . . . . . . . . . 15
4.5. Exercises 2B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6. 2.C Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.7. Exercises 2C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5. Chapter 3:Linear Maps 21
5.1. 3.A The vector space of linear maps . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2. Exercises 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3. 3.B Null Spaces and Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.4. Exercises 3B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.4.1. A way to construct (not) injective, surjective linear map . . . . . . . . . 25
5.4.2. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1
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5.5. Exercises 3C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.6. 3D: Invertibility and Isomorphic Vector Spaces . . . . . . . . . . . . . . . . . . . 31
5.7. Exercises 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.8. 3E: Products and Quotients of Vector Spaces . . . . . . . . . . . . . . . . . . . . 36
5.9. Exercises 3E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.10. 3F: Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.11. Exercises 3F. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6. Chapter 4:Polynomials 48
6.1. Exercise 4 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7. Chapter 5:Eigenvalues, Eigenvectors, and Invariant Subspaces 50
7.1. 5A: Invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2. Exercises 5A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3. 5B: Eigenvectors and Upper-Triangular Matrices . . . . . . . . . . . . . . . . . . 55
7.4. Exercises 5B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.5. 5C: Eigenspaces and Diagonal Matrices . . .. . . . . . . . . . . . . . . . . . . . . 58
7.6. Exercises 5C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8. Chapter 6:Inner Product Spaces 64
8.1. 6A: Inner Products and Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.2. Exercises 6A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.3. 6B: Orthonormal Bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.4. Exercises 6B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.5. 6C: Orthogonal Complements and Minimization Problems . . . . . . . . . . . . . 77
8.6. Exercises 6C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9. Chapter 7:Operators on Inner Product Spaces 81
9.1. 7A: Self-adjoint and Normal Operators . . .. . . . . . . . . . . . . . . . . . . . . 81
9.2. Exercises 7A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.3. 7B: The Spectral Theorem. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.4. Exercises 7B .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.5. 7C: Positive Operators and Isometries . . . . . . . . . . . . . . . . . . . . . . . . 88
9.6. Exercises 7C .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.7. 7D: Polar Decomposition and Singular Value Decomposition . . . . .. . . . . . . 91
9.8. Exercises 7D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.Chapter 8:Operators on Complex Vector Spaces 97
10.1. 8A: Generalized Eigenvectors and NilpotentOperators . . . . . . . . . . . . . . . 97
10.2. Exercises 8A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.3. 8B: Decomposition of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.4. Exercises 8B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10.5. 8C: Characteristic and Minimal Polynomials . . . .. . . . . . . . . . . . . . . . . 105
10.6. Exercises 8C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
, Toan Quang Pham page 3
10.7. 8D: Jordan Form .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
10.8. Exercises 8D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
11.Chepter 9:Operators on Real Vector Spaces 113
11.1. 9A: Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.2. Exercises 9A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.3. 9B: Operators on Real Inner Product Spaces . . . . . . . . . . . . . . . . . . . . 116
11.4. Exercises 9B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.Chapter 10:Trace and Determinant 119
12.1. 10A: Trace. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
12.2. Exercises 10A. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
13.Summary 122
14.Interesting problems 123
15.New knowledge 123
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