MA203/283 Semester 2 23/24

This is the homepage for Linear Algebra MA203/283 in the 23/24 academic year.

This course will run over the 12 weeks of Semester 2, with the first lecture on Wednesday January 10. It is an introduction to the concepts, methods and applications of linear algebra, for students specializing in mathematics and related subject areas.

## Welcome

You are very welcome to this course on linear algebra, which is running under both course codes MA203 and MA283, and has about 220 students. Linear algebra is a subject of fundamental importance in all areas of mathematics and its applications, and it is also a modern and exciting research area in its own right. It deals with the algebra of matrices, and also with central concepts such as dimension and bases. Its language and concepts are part of the basic roolkits of algebra, geometry, analysis, mathematical modelling and statistics. As well as providing essential background knowledge for all of these areas, it is a meeting point for all of them, full of topics that have a wide appeal.

If you are taking this course, you are already at a more advanced stage of mathematical education than most people in our world ever reach. Maybe you are excited about learning more and more advanced mathematics, or maybe you are taking this course to support your learning in another subject. Maybe you haven’t decided yet. You are welcome in all these cases. You are welcome if you have ever had a demoralizing experience in mathematical education, at any stage (or if you haven’t). You are welcome if you struggle with maths, or if you have yet to experience that inevitability. You are welcome if you belong to any group that is under-represented or marginalized, in mathematics or otherwise. This class aims for an inclusive and respectful working environment, please let the lecturer know if your experience here falls short of that.

Everyone is welcome. The following axioms (adapted from Federico Ardila) offer some scope for consideration.

- Mathematical potential is not static and is equally present in different groups.
- Everyone can have joyful, meaningful, and empowering mathematical experiences.
- Mathematics is a powerful, malleable tool that can be shaped and used for different purposes.
- Every person deserves to be treated with dignity and respect.

## Lectures and Content

The course content is organized into 20 “lectures”.

A “lecture” is a package of activity consisting of the following elements

- participation in the in-person lecture at the scheduled time and place;
- study of the accompanying
*slides*. The slides contain the*essential information*that students really need to understand, in order to complete the assessment and attain the learning outcomes to a satisfactory extent.

- study of the accompanying
- an accompanying section of the lecture notes. The lecture notes contain additional examples and more detailed explanations than the lecture slides, and facilitate a deeper level of engagement.
- Some exercises and tasks in the slides and notes.

- Solving a system of linear equations (January 10)
- How to present the solution to a linear system (January 12)
- Inconsistent Systems (and a computational tool) (January 17 and 19)
- Visualizing the solution to a linear system (January 24)
- Linear systems and intersecting hyperplanes (January 26)
- Matrix algebra (January 31)
- More matrix algebra: Matrix multipication (February 2)
- Gaussian elimination and matrix multiplication (February 7)
- Matrix algebra and linear transformations (February 9)
- Linear transformations and subspaces (February 14 and 16)
- Spanning sets (February 21)
- Linear Independence and Bases (February 23)
- Bases and Dimension (February 28)
- Change of Basis (March 1)
- Similarity (March 6)
- Eigenvectors and Diagonalizability (March 8)
- Algebraic and Geometric Multiplicity (March 13)
- Inner Product Spaces (March 15)
- Orthogonal Projection (March 20)
- Overdetermined systems and least squares approximation (March 22)

## Timetable

Wednesday 1.00–1.50, Anderson Lecture Theatre

Friday 11:00–11:50, Anderson Lecture Theatre

## Tutorials

Tutorials will commence in Week 3. The tutor is Deeba Javadpour. Times and venues are as follows:

Monday 2.00 ENG-2033

Monday 3.00 ENG-2033

Tuesday 5.00 (Tyndall lecture theatre)

Wednesday 2.00 AdB-G021

Everyone is welcome (and encouraged) to go to one weekly tutorial, according to their preference.

There will be three homework assignments on the Webwork system, each accounting for 10% of the module mark. The first will be posted early in the week of January 22. Anticipated due dates are February 9, March 1 and March 22. Each sheet will be posted for about three weeks.

We will also have practice problem sheets (below), for discussion in the tutorials and for independent study. The questions on the practice sheets connect to the Webwork assignments and also to the exam questions.

Problem Sheet 1 (posted January 19)

Problem Sheet 2 (posted February 7)

Problem Sheet 3 (posted March 3)

Problem Sheet 4 (posted March 19)

## Syllabus Outline

This course is an introduction to Linear Algebra, which is the study of *vector spaces*. Vector spaces are algebraic structures (sets) whose elements can be added, subtracted or multiplied by scalars (numbers). The algebra of matrices is closely related to linear algebra, and provides a concrete framework and a comprehensive computational environment for studying the subject. Much of our work will be concerned with matrices and their properties, but we will also be interested in the wider context of abstract vector spaces.

The concepts and methods of Linear Algebra is absolutely pervasive and have fundamental importance in every area of the mathematical and statistical sciences. Matrix algebra is a powerful tool in the mathematical modelling of physical and dynamic processes in both discrete and continuous contexts. Because linear algebra is relatively well understood and particularly amenable to computational methods, it has huge importance in scientific computing as well as in numerical analysis and approximation theory. It is also indispensable in geometry, combinatorics, analysis and all areas of algebra.

Our syllabus will have four chapters.

*Systems of linear equations*(Lectures 1-5)*Spanning sets, bases and dimension*(Lectures 6-10)*Linear Transformations, Eigenvectors and Similarity*(Lectures 11-15)

Choice of basis and change of basis. Diagonalizabilty. Eigenspaces.*Orthogonality, inner products and projections in vector spaces*(Lectures 16-20)

## Course and Lecture Notes

The lecture notes form the main text for the course. They will be updated (from last year) as we proceed, in line with this year’s edition of the module.

Chapter 1 Systems of linear equations |

Chapter 2 Spanning sets, bases and dimension |

Chapter 3 Linear transformations, eigenvectors and similarity |

Chapter 4 Inner products, orthogonality and projections |

**Assessment**

Assessment of students’ learning in MA283 will consist of the following elements.

- Three webwork
**homework assignments**(30%) - Two-hour
**final exam**in the summer exam session (70%) - Optional group or individual
**project**(20%)

More detail coming soon on the project element.

As you can see, this adds up to more than 100%. Total scores that exceed 100% will be returned as 100%.