# Linear Algebra

###  #### http://www.oup.co.uk/companion/singh #### Complete solutions to all exercises in the book are here:

Complete Solutions

The following video explains why linear algebra is important.

#### Chapter 1 Linear Equations and Matrices

Section 1.1

Systems of Linear Equations Pages 1-6.

Systems of Linear Equations 1.1 pages 6-11

Section 1.2

Section 1.2 Gaussian Elimination pages 12-15

Section 1.2 Gaussian Elimination pages 16-19

Section 1.2 Gaussian Elimination pages 19-22

Section 1.2 Gaussian Elimination pages 22-25

Section 1.2 Gaussian Elimination Examples

Exam question on Gaussian Elimination

Section 1.2 Reduced Row Echelon Form

Reduced Row echelon form for a larger system

Test on Gaussian Elimination – Section 1.2

Testing sections 1.1 and 1.2

Section 1.3

Chapter 1 Section 1.3 Vector Arithmetic pages 27-30

Chapter 1 Section 1.3 Vector Arithmetic pages 30-33

Chapter 1 Section 1.3 Vector Arithmetic pages 33-37

Section 1.4

Chapter 1.4 Arithmetic of Matrices pages 41-46

Chapter 1.4 Arithmetic of Matrices pages 52-55

Section 1.4 Arithmetic of Matrices Examples

Section 1.4 Arithmetic of Matrices Examples II

Test on Arithmetic of Matrices – Section 1.4

Practice test on addition, multiplication and solving equations

Section 1.5

Chapter 1.5 Matrix Algebra pages 52-62

Chapter 1.5 Matrix Powers page 70-73

Section 1.5 Matrix Algebra Examples

Section 1.5 Matrix Algebra another example

Section 1.5 Matrix Algebra more examples

Section 1.5 Online testing example

Section 1.4 and 1.5.pdf

Test on Manipulation of Matrices – Section 1.5

Section 1.6

Chapter 1.6 The Transpose and Inverse of a Matrix Pages 75-79

Chapter 1.6 The Identity and Inverse Matrix pages 80-82

Section 1.6 The Transpose and Inverse Matrix pages 83-85

Section 1.6 Properties of the Inverse Matrix pages 85-89

Section 1.6 Transpose and Inverse of a matrix examples

Section 1.6 Transpose and Inverse of a Matrix examples II

Proof based question on section 1.6

section 1.6 examples

Exam type questions on section 1.6

Section 1.7

Section 1.7 Types of Solutions pages 91-95

Section 1.7 Types of solutions example

Section 1.7.pdf

Exam type questions on section 1.7

Exam questions on sections 1.6 and 1.7.pdf

Section 1.8

Section 1.8 Inverse Matrix Method pages 105-108

Section 1.8 Inverse Matrix Method pages 108-110

Section 1.8 The Inverse Matrix Method pages 110-115

Section 1.8 Exercises 1.8 question 4(e) page 118

Section 1.8 Exercises 1.8 Question 4d page 118

Section 1.8 Exercises 1.8 question 4c page 118

Miscellaneous Exercises 1 Question 1.27 page 124

Exercises 1.8.pdf

Miscellaneous Exercises 1 Question 16 page 122

Test on chapter 1 of the book

#### Chapter 2 Euclidean Space

Section 2.1 Properties of Vectors pages 129-130

Section 2.1 Properties of Vectors pages 130-36

Section 2.1 Properties of Vectors pages 136-39

Exercises 2.1 Questions 3 and 5 page 142

Exam question 2.15 on pages 187-88

exercises 2.pdf

Test on vectors

Section 2.2 Inequalities pages 149-154

Section 2.2 Exercises 2.2 page 157

Exam question on Section 2.2

Notes for section 2,2 are here Section 2.2.pdf

Section 2.3 Linear Independence pages 159-165

Section 2.3 Linear Independence pages 165-69

Section 2.3 Linear Independence exercises 2.3

Section 2.3 Exam Question page 186

Notes for section 2.3 are here Section 2.3.pdf

#### Chapter 3 General Vector Spaces

Section 3.1 Introduction to Vector Spaces pages 191-194

Section 3.1 Introduction to Vector Spaces pages 196-200

Exercises 3.1 pages 201-202

Here are the notes for section 3.1 SECTION 3.1.pdf

Section 3.2 Subspace of a Vector Space pages 202-207

Section 3.2 Subspace of a Vector Space pages 208-209

Exercises 3.2 pages 214-16

Exercises 3.2 Questions 10 and 12 page 215

Exercises 3.2 part II pages 214-16

Here are the notes for section 3.2 SECTION 3.2.pdf

Section 3.3 Linear Independence and Basis pages 216-221

Section 3.3 Linear Independence and Basis pages 223-227

Exercises 3.3 Page 228

Exam questions on Section 3.3 page 272

Here are the notes for section 3.3  section 3.3.pdf

Section 3.4 Dimension pages 229-231

Exercises 3.4 pages 238-39

Exam questions on 3.4 page 274

Here are the notes for section 3.4  Section 3.4.pdf

Exercises 3.5 pages 252-53

Here are the notes for section 3.5  section 3.5.pdf

Section 3.6.1 Null Space pages 254-55

Section 3.6.1 Null Space pages 255-58

3.6.2 Properties of rank and nullity pages 259-67

Notes for section 3.6 are here Section 3.6.pdf

#### Chapter 4 Inner product spaces

Section 4.1 Introduction to I… Product Spaces pages 277-282

#### Chapter 6 Determinants and the inverse matrix

Section 6.1: Determinant of a 2 by 2 matrix Pages 431 – 438

Notes on the determinant of 2 by 2 matrix

Exercises 6.1 page 439

Here are the notes for section 6.1 section 6.1.pdf

Section 6.2: Cofactors Pages 441- 442

Section 6.2: Determinant and Inverse of a matrix Pages 442 to 452

Notes on Determinant of n by n and inverse matrix

Exercises 6.2 page 452

Here are the notes for Exercises 6.2 Sect ion 6.2.pdf

Exercises 6.2 page 453

Here are the notes for this part Section 6.2 part II.pdf

Test on Section 6.2

Test on Inverse and other properties of a matrix – Section 6.2

Exercises 6.3 page 470

Exercises 6.3 page 471

Here are the notes for this part

#### Chapter 7 Eigenvalues and eigenvectors

Section 7.1 Introduction to Eigenvalues and Eigenvectors Pages 491 497

Notes on Section 7.1 Pages 491-497

Section 7.1 Introduction to Evalues and Evectors Pages 497-502

Notes on Section 7.1 Pages 497 – 502

Online test on eigenvalues and eigenvectors

```Online test on eigenvalues and eigenvectors of 3 by 3 matrices

Test on Properties of Eigenvalues and Eigenvectors - Section 7.2```

Section 7.2 Properties of Eigenvalues\vectors Pages 503 – 507

Notes on Section 7.2 Pages 503 – 507

Section 7.2 Properties of Eigenvalues\vectors Pages 507 – 513

Notes on Section 7.2 Pages 507-513

Section 7.2 Cayley Hamilton Theorem pages 513-517

Notes on Section 7.2 Cayley Hamilton Theorem Pages 513-517

Section 7.3 Diagonalization pages 518-522

Notes on Section 7.3 Diagonalization Pages 518-522

Section 7.3 Introduction to Diagonalisation page 522 – 526

Exercises 7.3 question 8

Exercises 7.3 question 12

The notes for section 7.3 are here Section 7.3

Section 7.4 Orthogonal Diagonalization pages 537

Notes on Section 7.4 Orthogonal Diagonalization page 537

Exercises 7.4 question 4(a)

Exercises 7.4 question 5

Here are the notes for section 7.4 Section 7.4

Miscellaneous Exercises 7 question 21 exam question

Here are the notes for Miscellaneous exercises 7 Miscellaneous 7

A collection of 267 Supplementary Problems with Brief Solutions are at the following link: Supplementary Problems

A collection of 267 Supplementary Problems on Linear Algebra divided into chapters with complete solutions.

A collection of 60 Problems on Chapter 1

Complete Solutions to Problems on Chapter 1

A collection of 35 Problems on Chapter 2

Complete Solutions to Problems on Chapter 2

A collection of 41 Problems on Chapter 3

Complete Solutions to Problems on Chapter 3

A collection of 28 Problems on Chapter 4

Complete Solutions to Problems on Chapter 4

A collection of 27 Problems on chapter 5

Complete Solutions to Problems on chapter 5

A collection of 25 Problems on Chapter 6

Complete Solutions to Problems on Chapter 6

A collection of 51 Problems on Chapter 7

Complete Solutions to Problems on Chapter 7

Applications of Linear Algebra

SVD application by Becky Wheeler is here

An application of Linear algebra in quantum computing by Grant Hubbard is here

A good account of the history of mathematics is MacTutor History of Mathematics which is here.

Below is a set of supplementary notes on Linear Algebra.

Notes on Proof by Mathematical Induction.

Introductory CHAPTER:  Mathematical Logic, Proof, and Sets

 Section A: Joy of Sets Section A Exercise Ia Complete Solutions to Exercise Ia Section B: Subsets (Video)   Subsets notes Section B: Power Set Video Power set notes Proving results about Subsets (Video) Proving results Subsets SECTION B Subsets Exercise I  (b) Complete Solutions to Exercise I(b) Section C: Proposition Logic   ` ` Section I(c) Exercise I(c) Section D: Algebra of Propositions   Video of lecture Introductory: Algebra of Propositions Section D Section I(d) Exercise I(d) Complete solutions to Exercise I(d) Section E: Implication   Video of lecture Introductory Section E Implication Introductory Section E2 Tautology Tautology and Contradiction part I Tautology and Contradiction part II Sections E4 and E5: Converse and Contrapositive Section I(e)   On the following test enter your truth values in numerical order. `Test on Truth Tables` Exercise I(e) Complete solutions to Exercise I(e) Section F: Introduction to Proof   Section F1: If and only if Section F2: Introduction to Proof Section I(f) Exercise I(f) Complete solutions to Exercise I(f) Section G: Proofs   Section G2: Proof by Contrapositive Section G3: Without Loss of Generality SECTION I(g) PROOFS Exercise I(g) Section H: Proof by Contradiction   Section H: Proof by Contradiction H1: Introduction to Proof by Contradiction Section H2: Examples of Proof by Contradiction SECTION I(h) PROOF  BY CONTRADICTION Exercise I(h) Complete Solutions to Exercise I(h) Section I: Principle of Mathematical Induction Section I(i) Exercise I(i) Complete Solutions to Exercise I(i)

Complete Solutions to Exercise 1j

Supplementary Problems on Mathematical Induction

Complete Solutions to Supplementary Problems on Mathematical Induction

#### Cramer Rule

SECTION E-Applications of Determinant

Exercise 6e

Complete Solutions to Exercise 6e

#### Chapter 7: Eigenvalues and Eigenvectors

Complete solutions to chapter 7

Section 7g Sketching Conics Harish Chandra 1923 to 1983

In mathematics, there is an empty canvas before you, which can be filled without reference to external reality.

Mathematicians on Creativity by Peter Borwein Hermann Schubert (1848 to 1911)

The three positive characteristics that distinguish mathematical knowledge from other knowledge may be briefly expressed as follows; first, mathematical knowledge bears more distinctly the imprint of truth on all its results than any other kind of knowledge; secondly, it is always a sure preliminary step to the attainment of other correct knowledge; thirdly, it has no need of other knowledge.

Hermann Schubert in ‘Mathematical Essays and Recreations’ 1898

This is a direct quote from ‘Mathematicians on Creativity by Peter Borwein

#### Test and Examination Papers in Linear Algebra

A good way to revise for examination is to try past examination papers. It is important that you attempt the paper without looking at the solutions. Even if you get stuck, it is better to have a go at the question and then at the end mark your paper. This is a much more effective way of learning because you will have an understanding of the material and now what is required of you. Of course, this is much harder but also more rewarding.