# MA106 Linear Algebra

**Lecturer: **Diane Maclagan and Marco Schlichting

**Term(s):** Term 2

**Status for Mathematics students:** Core for Maths

**Commitment:** 30 one-hour lectures

**Assessment:** 15% from weekly assignments, 85% from a 2 hour examination

**Formal registration prerequisites: **None

**Assumed knowledge: **A-level Mathematics and Further Mathematics

**Useful background:** A-level Mathematics and Further Mathematics

**Synergies: **All parts of mathematics, and more generally, all parts of quantitative science, use linear algebra

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA241 Combinatorics
- MA243 Geometry
- MA251 Algebra I: Advanced Linear Algebra
- MA254 Theory of ODEs
- MA258 Mathematical Analysis III
- MA259 Multivariable Calculus
- MA3D5 Galois Theory
- MA3E1 Groups and Representations
- MA398 Matrix Analysis and Algorithms
- MA3K1 Mathematics of Machine Learning
- MA3H7 Control Theory
- MA427 Ergodic Theory
- MA4J1 Continuum Mechanics

**Content**: Many problems in maths and science are solved by reduction to a system of simultaneous linear equations in a number of variables. Even for problems which cannot be solved in this way, it is often possible to obtain an approximate solution by solving a system of simultaneous linear equations, giving the "best possible linear approximation''.

The branch of maths treating simultaneous linear equations is called linear algebra. The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another. It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix.

These theoretical ideas have many applications, which will be discussed in the module. These applications include:

- Solutions of simultaneous linear equations
- Properties of vectors
- Properties of matrices, such as rank, row reduction, eigenvalues and eigenvectors
- Properties of determinants and ways of calculating them

**Aims**: To provide a working understanding of matrices and vector spaces for later modules to build on and to teach students practical techniques and algorithms for fundamental matrix operations and solving linear equations.

**Objectives**: Students must understand the ideas of linearly independent vectors, spanning sets and bases of vector spaces. They must also understand the equivalence of linear maps between vector spaces and matrices and be able to row reduce a matrix, compute its rank and solve systems of linear equations. The definition of a determinant in all dimensions will be given in detail, together with applications and techniques for calculating determinants. Students must know the definition of the eigenvalues and eigenvectors of a linear map or matrix, and know how to calculate them.

**Books**:

David Towers, *Guide to Linear Algebra*, Macmillan 1988.

Howard Anton, *Elementary Linear Algebra*, John Wiley and Sons, 1994.

Paul Halmos, *Linear Algebra Problem Book*, MAA, 1995.

G Strang, *Linear Algebra and its Applications*, 3rd ed, Harcourt Brace, 1988.