# Linear Algebra:  2022-2023

 Lecturer Stefan Kiefer Degrees Term Michaelmas Term 2022  (20 lectures)

## Overview

The course introduces 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 in the academic year 2019-2020 and then again in 2021-2022.

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:

• Solve linear systems of equations.
• Comprehend vector spaces and subspaces.
• Understand fundamental concepts of linear maps, including isomorphisms, range and nullspace, matrices, change of bases and projections.
• Deal with Determinants.
• Understand similarity, including eigenvalues, eigenvectors, and diagonalisation.
• 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. They follow closely the corresponding chapters from the textbook.

• Lectures 1-3 Linear Systems: solving linear systems; linear geometry; reduced echelon form.

• Lectures 4-7 Vector Spaces: definition; linear independence; basis and dimension.

• Lectures 8-13 Maps Between Spaces: isomorphisms; homomorphisms; computing linear maps; matrix operations; change of basis; projection.

• Lectures 14-15 Determinants: definition; geometry of determinants; Laplace's formula.

• Lectures 16-17 Similarity: definition; eigenvectors and eigenvalues.
• Lectures 18-20 Least Squares and Factorisations: least squares; LU factorisation; QR factorisation; singular value decomposition.

## Syllabus

• Linear systems
• Reduced echelon form
• Vector spaces
• Linear independence
• Basis and dimension
• Linear maps
• Isomorphism
• Range and null space
• Matrices
• Change of basis
• Orthogonal projection
• Determinants
• Laplace's formula
• Similarity
• Eigenvalues and eigenvectors
• Diagonalisation
• Least squares (CS Prelims only)
• LU factorisation, QR factorisation, singular value decomposition (CS Prelims only)