# Linear Algebra

This page is dedicated to Shaheed Bibi Paramjit Kaur.

#### Linear Algebra Step by Step – the URL of the book is

#### http://www.oup.co.uk/companion/singh

#### Complete solutions to all exercises in the book are here:

The following video explains why linear algebra is important.

https://www.youtube.com/watch?v=kjBOesZCoqc&index=1&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

#### Topic Videos and Test Questions

**All the test questions below were developed by Dr Martin Greenhow and his team at Brunel University London. (You can enter any name and any number like 123 for your student number. Also don’t need to give your email address, just enter a school where it asks for your email address.) You can do the test as many times as you wish as the numbers change every time you refresh the page. **

#### 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 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**

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 47-52

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

**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

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

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

Miscellaneous Exercises 1 Question 16 page 122

#### 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

Section 2.2 Further Properties of Vectors pages 143-49

Section 2.2 Inequalities pages 149-154

Section 2.2 Exercises 2.2 page 157

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

Section 2.4 Basis and Spanning Set pages 178-182

#### 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

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 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

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

Exam questions on 3.4 page 274

Here are the notes for section 3.4 Section 3.4.pdf

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…r Product Spaces pages 282-85

Section 4.1 Introduction to I…r Product Spaces pages 286-88

Exercises 4.1 Inner Product Space pages 288-90

Exam question on Inner Product Spaces pages 335-36

Notes on section 4.1 are here section 4.1.pdf

#### 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

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

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

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

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

Here are the notes for this part section 6.3.pdf

#### 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

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

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**

Complete Solutions to Exercise 1j

Supplementary Problems on Mathematical Induction

Complete Solutions to Supplementary Problems on Mathematical Induction

#### Chapter 6: Determinants

#### Cramer Rule

SECTION E-Applications of Determinant

Complete Solutions to Exercise 6e

#### Chapter 7: Eigenvalues and Eigenvectors

Complete solutions to chapter 7

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.

#### Linear Algebra web link:

The following is a fantastic source of videos on linear algebra. Please check it out.