# Linear Algebra:  2017-2018

 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 1-3  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 4-6 Independence and orthogonality:  Linear independence of vectors.  Basis and dimension of a vector space.  Orthogonal vectors and subspaces.  The Gram-Schmidt orthogonalisation, related algorithms and operation count. Application examples.

• Lectures 7-9  Matrices:  Matrix operations.  Column, row and null space.  Rank of a matrix.  Inverse and transpose.  Elementary matrices.  The Gauss-Jordan 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 11-12  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 13-14  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 15-17 Linear transformations:  Definition and examples. Properties and Composition of linear transformations. Rotations, reflections and stretches. Translations using homogeneous coordinates. One-to-one and onto transformations.

• Lectures 18-20 Eigenvalues and eigenvectors:  Definition. Similarity and diagonalization. Population growth revisited. Systems of linear differential equations.
• Lectures 21-22 Iterative methods for solving linear equations:  The methods of Jacobi, Gauss-Seidel, successive over relaxation, and steepest descent.  Error analysis.
• Lectures 23-24 Over determined systems:  The QR factorisation, least squares problems.
• ## Syllabus

• Vector spaces and subspaces
• Matrices
• Inverse matrices
• Solution of linear systems
• Elementary matrix factorisations
• Iterative methods
• Linear transformations
• Eigenvalues and eigenvectors