## MATH2200 Linear Algebra 2

### 10 creditsClass Size: 200

Module manager: Dr V. Kisil
Email: kisilv@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisites

 MATH1015 Linear Algebra 1

### This module is mutually exclusive with

 MATH2080 Further Linear Algebra

This module is approved as an Elective

### Module summary

This module carries on from Linear Algebra 1 (MATH 1015) and develops the concept of a linear transformation on an abstract vector space. These ideas are then applied to questions about 'changing variables' so as to obtain the simplest form of a matrix or a quadratic form.

### Objectives

On completion of this module, students should be able to:
a) handle elementary arguments with linear independence, spanning, dimension, and sums of vector spaces over real, complex and finite fields;
b) represent a linear transformation by a matrix with respect to a given basis;
c) determine whether a matrix is diagonalisable and calculate its invariants such as the minimum polynomial;
d) perform standard calculations in real inner product spaces, including the Gram-Schmidt process;
e) diagonalise a quadratic form and determine its rank and signature;
f) show a grasp of the underlying theory by proving the main theorems in the course, and finding other elementary proofs.

### Syllabus

1. Vector spaces, sums of subspaces, over the reals, complexes, or finite fields.
2. Linear transformations and representation of a linear transformation by a matrix. The AP = PB theorem.
3. Diagonalisation of a matrix. Cayley-Hamilton theorem and the minimum polynomial of a matrix. Jordan normal form.
4. Inner product and Euclidean spaces, orthogonal vectors and the Gram-Schmidt process.
5. Quadratic forms and diagonalisation of real and symmetric matrices.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Example Class 11 1.00 11.00 Lecture 22 1.00 22.00 Private study hours 67.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 15.00 Total percentage (Assessment Coursework) 15.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated