### 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:

**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. **

There are also test questions from Newcastle University using Numbas and these were developed by Bill Foster and Christian Perfect.

Notes on Mathematical Induction

Introductory D Principle of Mathematical Induction

Complete Solutions to Exercise I(d)

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**

##### 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**

##### 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

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 (Links to an external site.)

###### 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

### 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 Further Properties of Vectors pages 143-49

###### 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

###### Section 2.4 Basis and Spanning Set pages 178-182

###### Exercises 2.4 pages 183-85

###### Exercises 2.4 part II pages 183-85

###### Here are the notes for section 2.1 SECTION 2.4.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

###### Section 4.1 Introduction to I…r Product Spaces pages 282-85

#### chapter 6 Determinants and the inverse matrix

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

**Test on Section 6.2**

Test on Inverse and other properties of a matrix - Section 6.2Test on Sections 6.2 and 6.3

#### 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 matricesTest 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

Section 7.4 Orthogonal Diagonalization pages 537

Notes on Section 7.4 Orthogonal Diagonalization page 537

#### Supplementary Problems with Brief Solutions are at the following link:

If you are an academic and would like complete solutions to these problems then email me at k.singh@herts.ac.uk

Challenging Problems on Linear Algebra with complete solutions are Here

### Applications of Linear Algebra

SVD application by Becky Wheeler 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

Demonstration of eigenvalue and eigenvector here

#### 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)

â€œâ€¦ the three positive characteristics that distinguish mathematical *knowledge* from other knowledge â€¦ 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.â€

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 links:

The following video explains why linear algebra is important.

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

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