### 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

#### Amazon Reviews of Linear Algebra Step by Step are here

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

### Videos Topics

Notes on Mathematical Induction

Introductory D Principle of Mathematical Induction

Complete Solutions to Exercise I(d)

Chapter 1 Linear Equations and Matrices

###### Systems of Linear Equations Pages 1-6.

###### Systems of Linear Equations 1.1 pages 6-11

###### 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

###### 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

###### 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

###### Chapter 1.5 Matrix Algebra pages 52-62

###### Chapter 1.5 Matrix Powers page 70-73

###### 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 1.7 Types of Solutions pages 91-95

###### 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

###### 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

###### Section 2.2 Further Properties of Vectors pages 143-49

###### Section 2.2 Inequalities pages 149-154

###### Section 2.3 Linear Independence pages 159-165

###### Section 2.3 Linear Independenc pages 165-69

### 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

###### Section 3.2 Subspace of a Vector Space pages 202-207

###### Section 3.2 Subspace of a Vector Space pages 208-209

###### Section 3.3 Linear Independence and Basis pages 216-221

###### Section 3.3 Linear Independence and Basis pages 223-227

###### Section 3.4 Dimension pages 229-231

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

Section 6.2: Cofactors Pages 441- 442

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

Notes on Determinant of n by n and inverse matrix

Test 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**

(These online tests were developed by Dr Martin Greenhow and his team at Brunel University London.)

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 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.