Exam (elaborations)
MAT2611 linear_al_done_right_notes.
- Course
- MAT2611 - Linear Algebra
- Institution
- University Of South Africa
MAT2611 linear_al_done_right_notes.100% CORRECT questions, answers, workings and explanations. for assistance.
[Show more]
Content preview
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
,Toan Quang Pham page 2
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller LOVELY01. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R51,78. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75323 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy summaries for 14 years now