Linear Algebra: 20162017
Lecturer 

Degrees 

Term 
Michaelmas Term 2016 (24 lectures) 
Overview
This is a first course in linear algebra. The course will lay down basic concepts and techniques of linear algebra, and provide an appreciation of the wide application of this discipline within the scientific field.The course will require development of theoretical results. Proofs and consequences of such results will require the use of mathematical rigour, algebraic manipulation, geometry and numerical algorithms.
The course will provide insight into how linear algebra theorems and results, sometimes quite abstract, impinge on everyday life. This will be illustrated with detailed examples.
Learning outcomes
At the end of this course the student will be able to:
 Comprehend vector spaces and subspaces.
 Understand fundamental properties of matrices including determinants, inverse matrices, matrix factorisations, eigenvalues and linear transformations.
 Solve linear systems of equations.
 Have an insight into the applicability of linear algebra.
Synopsis

Lectures 13 Vectors: Vectors and geometry in two and three space dimensions. Algebraic properties. Dot products and the norm of a vector. Important inequalities. Vector spaces, subspaces and vector space axioms. Application examples.

Lectures 46 Independence and orthogonality: Linear independence of vectors. Basis and dimension of a vector space. Orthogonal vectors and subspaces. The GramSchmidt algorithm.

Lectures 79 Matrices: Column and row space. Range and null space. Rank of a matrix. Matrix operations. Determinant and inverse. Elementary matrices. Application examples: population growth and finite linear games.

Lectures 1012 Systems of linear equations: Examples of linear systems. Gaussian elimination and pivoting. Row echelon form. Elementary matrix factorisations. Application examples: network analysis and global positioning systems. Problems arising from solving linear equations on a computer.

Lectures 1315 Linear transformations: Definition and examples. Properties and Composition of linear transformations. Rotations, reflections and stretches. Translations using homogeneous coordinates. Onetoone and onto transformations.
 Lectures 1619 Eigenvalues and eigenvectors: Definition. Similarity and diagonalization. Population growth revisited. Systems of linear differential equations.
 Lectures 2021 Iterative methods for solving linear equations: The methods of Jacobi, GaussSeidel, successive over relaxation, and steepest descent. Error analysis.
Syllabus
 Vector spaces and subspaces
 Matrices
 Inverse matrices
 Iterative methods
 Linear transformations
 Solution of linear systems
 Eigenvalues and eigenvectors
Reading list
 Introduction to linear algebra (3rd Edition), Gilbert Strang, WellesleyCambridge press.