Introduction to linear equations and matrices

Introduction

Please note: a Statement of Participation is not issued for this course.

This free OpenLearn course, Introduction to linear equations and matrices, is an extract from the Open University course M208 Pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a second level course that introduces the three main branches of pure mathematics, namely group theory, analysis and linear algebra. Proofs are a vital part of pure mathematics. Students studying M208 are expected to read through a number of proofs to improve understanding of the course material and to develop mathematical skills, including producing convincing arguments and problem solving.

Introduction to linear equations and matrices consists of material from M208 Unit LA2, Linear equations and matrices, and has five sections in total. You should set aside between three to four hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 5%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

In this extract, matrices are used as a concise way of representing systems of linear equations. It is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the course. However, a few techniques and definitions taught in earlier units in M208 are present in the extract without explanation. 

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M208 video material (although please note that the PDF may contain references to other parts of M208). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 looks at simultaneous linear equations in two and three unknowns. The idea of a solution set is introduced, and results are interpreted geometrically. These ideas are then generalised to systems of linear equations and you are introduced to the Gauss-Jordan elimination method of solution.

Section 2 develops a strategy for solving systems of linear equations.

Section 3 looks at the algebra of matrices. Addition and scalar multiplication are investigated, and you will see that matrices can be thought of as a generalisation of vectors. The dot product of vectors is used to define multiplication of matrices. Towards the end of the section the matrix operation of transposition and some important types of matrices are introduced.

Section 4 introduces the inverse of a matrix and gives you a method for finding an inverse when it does exist. The idea of invertibility of a matrix is linked with the number of solutions of some systems of linear equations.

Section 5 uses systems of linear equations to introduce the determinant of a square matrix.

This OpenLearn course is an adapted extract from the Open University course M208 Pure mathematics.