# Mathematics for Computer Science and Philosophy:  2019-2020

 Degrees

## Overview

This is a second/third-year course for Computer Science & Philosophy students. They will attend a subset of the first-year lectures in Linear Algebra and Continuous Mathematics, and will so be equipped with the knowledge needed for courses in Machine Learning.

Classes:

Colleges will arrange problem classes for their students, as with the corresponding first-year courses: three classes for Linear Algebra and two for Continuous Mathematics.

Examinations:

A two-hour exam at the end of the third year, with students answering two questions from a choice of three.

## Synopsis

Linear Algebra. About 14 lectures in Michaelmas Term.

• Complex numbers.
• 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. Application examples.
• 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.

• 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.
• 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 examples: network analysis, global positioning systems and intersection of planes.
• 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; equa­tions of lines and planes.
• Eigenvalues and eigenvectors: Definition. Similarity and diagonaliza-tion. Population growth revisited. Systems of linear differential equa­tions.

Continuous Mathematics. About 9 lectures in Hilary Term.

• Derivatives, partial derivatives, differentiation with respect to a vector, gradient, Hessian and Jacobian. Taylor’s theorem in 1 dimension (La­grange remainder), Taylor’s theorem in n dimensions (remainder only briefly). Examples.
• Optimization in 1 and n Classification of turning points via Taylor’s theorem. Convexity. Constrained optimization: Lagrange multipliers. Examples.
• Convergence rates of iterative methods.
• Numerical root finding in 1 dimension by bisection, Newton’s method, secant method. Root finding in n dimensions by Newton’s method, and briefly quasi-Newton methods. Complexity and error analysis. Examples.

## Syllabus

Vector spaces and subspaces. Matrices. Inverse matrices. So­lution of linear systems. Elementary matrix factorisations. Eigenvalues and eigenvectors.

Derivatives, partial derivatives, differentiation with respect to a vector; gradient, Hessian, Jacobian. Taylor’s theorem in 1- and n-dimensions, with Lagrange remainder. Optimization in 1- and n-dimensions, classification of stationary points. Constrained optimization and the method of Lagrange multipliers. Iterative numerical methods and rates of convergence. Methods for numerical root finding in 1- and n-dimensions, complexity and (simple cases of) error analysis.