MA10612 Linear Algebra
Introductory description
Many problems in maths and science are solved by reduction to a system of simultaneous linear equations in a number of variables. Even for problems which cannot be solved in this way, it is often possible to obtain an approximate solution by solving a system of simultaneous linear equations, giving the "best possible linear approximation''.
The branch of maths treating simultaneous linear equations is called linear algebra. The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another. It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix.
These theoretical ideas have many applications, which will be discussed in the module. These applications include:
Solutions of simultaneous linear equations.
Properties of vectors.
Properties of matrices, such as rank, row reduction, eigenvalues and eigenvectors.
Properties of determinants and ways of calculating them.
Module aims
To provide a working understanding of matrices and vector spaces for later modules to build on and to teach students practical techniques and algorithms for fundamental matrix operations and solving linear equations.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 The vector space R^n, including a geometric description of vector addition in R^2.
 Fields. Definition of a vector space V over a field. The space spanned by a subset of V. Linear dependence and independence. Bases. Dimension. Subspaces. Dual spaces and dual bases.
 Linear maps f:V>W. Isomorphism of vector spaces. Any ndimensional vector space over F is isomorphic to R^n. Examples of linear maps, including differentiation and integration as linear maps on spaces of functions or polynomials.
 Matrices. Algebraic operations on matrices. Reduction of a matrix using row and column operations. Application to the solution of linear equations. Rank. Row rank = Column rank.
 The relation between linear maps and matrices. the matrix of a linear map with respect to a given basis. Change of basis changes A to PAQ^{1}. The kernal and image of f:V>W. The rank and nullity of f.
 Determinants, defined by ∑σϵSn sign σ(∏ai,σ(i)). Det(AB) = Det(A)Det(B) (proof either in general or in the cases n=1,2,3). Submatrices, minors, cofactors, the adjoint matrix. Rules for calculating determinants. The inverse of a matrix. Ax=0 has nonzero solution if and only if det(A)=0. Determinantal rank.
 Eigenvalues and eigenvectors. Definition and examples. Their geometric significance. Diagonalisation of matrices with distinct eigenvalues.
 Inner product spaces and isometries. Euclidean spaces. Orthogonal transformations and matrices.
Learning outcomes
By the end of the module, students should be able to:
 Understand and demonstrate knowledge of vector spaces, fields, linear dependence and independence, bases and dimension.
 Understand linear transformations and be able to show examples of linear maps such as differentiation and integration as linear maps on spaces of functions of polynomials.
 Be proficient at matrix manipulation, reduction of a matrix using row and column operations and be able to apply to finding solutions to linear equations.
 Be able to compute determinants for general n by n matrices, compute cofactors and adjoint matrices and understand the implications of doing this to solving sets of linear equations.
 Be able to compute eigenvalues and eigenvectors of matrices and understand their geometric significance. Be able to diagonalize matrices with distinct eigenvalues.
Indicative reading list
David Towers, Guide to Linear Algebra, Macmillan 1988.
Howard Anton, Elementary Linear Algebra, John Wiley and Sons, 1994.
Paul Halmos, Linear Algebra Problem Book, MAA, 1995.
G Strang, Linear Algebra and its Applications, 3rd ed, Harcourt Brace, 1988.
Subject specific skills
To provide a working understanding of matrices and vector spaces for later modules to build on and to teach students practical techniques and algorithms for fundamental matrix operations and solving linear equations.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (25%) 
Tutorials  6 sessions of 30 minutes (2%) 
Private study  87 hours (72%) 
Total  120 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D2
Weighting  Study time  

Assessment (nonMaths students)  15%  
weekly, summative, assignments 

2 hour online examination (Summer)  85%  
Exam

Assessment group R
Weighting  Study time  

2 hour examination  100%  
exam

Feedback on assessment
Marked assignments, face to face supervisions.
Courses
This module is Core for:
 Year 1 of USTAG302 Undergraduate Data Science
 Year 1 of USTAG304 Undergraduate Data Science (MSci)
 Year 1 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 1 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 1 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 1 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 1 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 1 of UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 1 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 1 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 1 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 1 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 1 of UMAAGV17 Undergraduate Mathematics and Philosophy
 Year 1 of UMAAGV18 Undergraduate Mathematics and Philosophy with Intercalated Year
 Year 1 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 1 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 1 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 1 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 1 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 1 of UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 1 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics