# Linear Algebra:  2019-2020

 Lecturer Jonathan Whiteley Degrees Term Michaelmas Term 2019  (20 lectures)

## Overview

The course will introduce basic concepts and techniques from linear algebra that will be required in later courses in areas such as machine learning, computer graphics, quantum computing. The theoretical results covered in this course will be proved using mathematically rigorous proofs, and illustrated using suitable examples.

The syllabus for the Preliminary Examination in Computer Science changed with effect from the academic year 2019-2020. Material on iterative solution to linear equations and least squares solutions of over-determined systems has been removed. Past exam questions on these topics are therefore not suitable when attempting past exam questions.

This course is part of both the Preliminary Examination for Computer Science students and the Final Honour School for Computer Science and Philosophy students. Questions set from this course for the Final Honour School in Computer Science and Philosophy will be more challenging than those that are set for the Preliminary Examination in Computer Science, and students taking this exam should bear this in mind when attempting sample exam questions and past exam questions.

## 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 1-20 cover the syllabus for the Preliminary Examination in Computer Science.

Lectures 1-17 cover the syllabus for the Final Honour School in Computer Science and Philosophy.

• Lectures 1-2  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.  Complex vector spaces.

• Lectures 3-5 Independence and orthogonality: Linear independence of vectors.  Basis and dimension of a vector space.  Orthogonal vectors and subspaces.  The Gram-Schmidt orthogonalisation.

• Lectures 6-8  Matrices:  Matrix operations.  Column, row and null space.  Rank of a matrix.  Inverse and transpose.  Elementary matrices.  The Gauss-Jordan method.

• Lectures 9-11  Systems of linear equations:  Examples of linear systems.  Geometry of linear equations.  Gaussian elimination.  Row echelon form.  Homogeneous and nonhomogeneous systems of linear equations.  Application to the intersection of lines and planes.

• Lectures 12-14  Elementary matrix factorisations and determinants: LU factorisation, related algorithms and operation count.  PLU factorisation.  Calculating the determinant of a matrix.  Properties of the determinant of a matrix.  Application examples: area, volume and cross product.
• Lectures 15-17  Eigenvalues and eigenvectors: Definition.  Similarity and diagonalisation.
• Lectures 18-20  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.

## Syllabus

• Vector spaces and subspaces
• Linear independence and bases for vector spaces
• Orthogonal vector spaces and the Gram-Schmidt orthogonalisation process
• Matrices
• Inverse matrices
• Solution of linear systems
• Elementary matrix factorisations
• Determinants of matrices
• Eigenvalues and eigenvectors
• Linear transformations (CS Prelims only)