Linear Algebra

 Linear Algebra

This page is dedicated to Shaheed Bibi Paramjit Kaur.

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

9780199654444_4501

Complete solutions to all exercises in the book are here:

Complete Solutions

Test

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. 

This is not Dr Martin Greenhow but one of his dogs.

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

Proof by Induction

Proof by Induction Part II

Notes on Mathematical Induction

Introductory D Principle of Mathematical Induction

Exercise I(d)

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 

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 (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

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

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 6.2

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

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

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:

Supplementary Problems

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

Section A: Joy of Sets

 

Section A Introduction to Joy of Sets video

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


Chapter 6: Determinants

Cramer Rule

SECTION E-Applications of Determinant

Exercise 6e

Complete Solutions to Exercise 6e

Chapter 7: Eigenvalues and Eigenvectors

Demonstration of eigenvalue and eigenvector here

Complete solutions to chapter 7

Section 7f Quadratic Forms

Section 7g Sketching Conics


harish-chandra

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

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 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.

Linear Algebra at MIT in the USA