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 6.2 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:
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)
“… 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.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.