# Linear Algebra

###  #### http://www.oup.co.uk/companion/singh #### Complete solutions to all exercises in the book are here:

Complete Solutions

Test

### Topic Videos and Test Questions

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 1.6
###### 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

### Chapter 3 General Vector Spaces

#### 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

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 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

#### 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 7g Sketching Conics 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.