Linear Algebra: 20172018
Lecturer 

Degrees 

Term 
Michaelmas Term 2017 (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 orthogonalisation, related algorithms and operation count. Application examples.

Lectures 79 Matrices: Matrix operations. Column, row and null space. Rank of a matrix. Inverse and transpose. Elementary matrices. The GaussJordan method. Application examples: population growth and finite linear games.

Lecture 10 Systems of linear equations: Examples of linear systems. Geometry of linear equations. Gaussian elimination. Row echelon form. Homogeneous and nonhomogeneous systems of linear equations.

Lectures 1112 Elementary matrix factorisations. LU factorisation, related algorithms and operation count. PLU factorisation. Solving systems of linear equations. Application examples: network analysis, global positioning systems and intersection of planes.
 Lectures 1314 Determinants. Calculating the determinant of a matrix. Properties of the determinant of a matrix. Application examples: area, volume and cross product; equations of lines and planes.

Lectures 1517 Linear transformations: Definition and examples. Properties and Composition of linear transformations. Rotations, reflections and stretches. Translations using homogeneous coordinates. Onetoone and onto transformations.
 Lectures 1820 Eigenvalues and eigenvectors: Definition. Similarity and diagonalization. Population growth revisited. Systems of linear differential equations.
 Lectures 2122 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
 Solution of linear systems
 Elementary matrix factorisations
 Iterative methods
 Linear transformations
 Eigenvalues and eigenvectors
Reading list
 Introduction to Linear Algebra, Gilbert Strang, WellesleyCambridge press.