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The University of Southampton

MATH1048 Linear Algebra I

Module Overview

Linear maps on vector spaces are the basis for a large area of mathematics, in particular linear equations and linear differential equations, which form the basic language of the physical sciences. This module restricts itself to the vector space R^n to build an intuitive understanding of the concepts of linear algebra and tools for calculations. We begin with the geometry of lines and planes in R^3 and R^n looking at the intuitive concept of vectors on the one hand, and with systems of linear equations on the other. This leads us to matrix algebra, and in particular the inversion of matrices. One of the pre-requisites for MATH1049, MATH1057, MATH1058, MATH1060, MATH2013, MATH2045, MATH3087, MATH3033 and MATH3090

Aims and Objectives

Module Aims

This module restricts itself to the vector space R^n to build an intuitive understanding of the concepts of linear algebra and tools for calculations.

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • Solve systems of linear equations
  • Calculate the determinants, and invert matrices
  • Apply linear algebra methods to geometric problems in R^3 and R^n.


• Complex arithmetic • Vectors in R^n: examples from R³, equations of lines and planes in R³. • Systems of linear equations, Gaussian elimination. • Matrix algebra: nxm matrices, sums, products, transpose, inverse of an nxn matrix, matrix equations. • Determinants, cofactor definition of the inverse, proof that detA≠O if and only if A invertible. • Properties of R^n, subspaces of R^n, span, null space and column space of a matrix. • Linear independence of vectors in R^n. • Scalar product and geometrical applications. • Vector product and applications in R³. • Linear transformations in R^n, examples in R² and R³. • Eigenvalues and eigenvectors. Throughout the module some theorems will be proved.

Learning and Teaching

Independent Study96
Total study time150

Resources & Reading list

Larson, Edwards, Falvo. Elementary Linear Algebra. 

Lang Serge. Introduction to Linear Algebra. 

Lay David C -. Linear Algebra and its applications. 

Hirst & Singerman (2000). Basic Algebra and Geometry. 

Martin Anthony and Michele Harvey (2012). Linear Algebra: Concepts and Methods. 

Edwards CH & Penney DE. Elementary Linear Algebra. 

Leon Steven J. Linear Algebra with Application. 



MethodPercentage contribution
Class Test 10%
Coursework 20%
Exam  (120 minutes) 70%


MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Sudents will be provided with full lecture notes. Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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