The University of Southampton
Courses

# 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

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Apply Linear Algebra methods to geometric problems in R^3 and R^n.
• Solve systems of linear equations and apply this to other questions from Linear Algebra
• Calculate the determinants, invert and perform basic operations with matrices
• Work with linear transformations of R^n and their matrices.
• Find eigenvalues/eigenvectors of square matrices; diagonalize symmetric matrices

### Syllabus

- 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

#### Teaching and learning methods

Lectures, problem classes and workshops

TypeHours
Independent Study96
Teaching54
Total study time150

Larson, Edwards, Falvo. Elementary Linear Algebra.

Lay David C -. Linear Algebra and its applications.

Leon Steven J. Linear Algebra with Application.

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

Lang Serge. Introduction to Linear Algebra.

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

Edwards CH & Penney DE. Elementary Linear Algebra.

### Assessment

#### Summative

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

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

### Costs

#### 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 www.calendar.soton.ac.uk.