MT1212 Linear Algebra
Course Facts
- Lecturer:
Prof. Alexandre Borovik , Room N115 MSS Building, Extension 63645 - Delivery: Semester Two
- Credits: 15
- Lectures: see timetable
- Prerequisites: A-Level Mathematics or equivalent
- Teaching and learning methods: 3 lectures and 1 supervision per week
- Assessment:
Page Contents
Course Description
Syllabus
- Systems of linear equations: coefficient matrix and augmented matrix, elementary row operations, row echelon form and reduced row echelon form, solution of systems using Gauss and Gauss-Jordan algorithms. [Poole 2.1, 2.2, 2.3; 4 Lectures]
- Algebra of matrices: addition, multiplication, inverse, transpose, elementary matrices and elementary row operations, row equivalence, calculation of inverses by Gauss-Jordan algorithm, finding non singular P so that PA is in (reduced) row echelon form, row canonical form. [Poole 3.1, 3.2, 3.3, 3.4, 3.7; 6 Lectures]
- Subspaces of Rn: the null space and column space of a matrix A and their significance for Ax = b, linear independence, spanning sets, basis of a subspace, dimension of a subspace, using Gauss algorithm to find a basis for null space, column space and row space, nullity and rank of an m x n matrix, nullity + rank = n, row rank = column rank, coordinates and change of coordinates. [Poole 3.5; 6 Lectures]
- Standard inner product on Rn: standard inner product, length of vector, orthogonal vectors, orthonormal basis, coordinates with respect to an orthonormal basis, Gram-Schmidt algorithm, null space of A is orthogonal complement of row space. [Poole 5.1, 5.2, 5.3; 3 Lectures]
- Vector spaces over a field R: definition and examples (including function spaces, spaces of matrices), basis and dimension, basis, dimension, coordinates, change of coordinates, subspaces, intersections, sum and direct sum, formula for dim(U + V). [Poole 6.1, 6.2, 6.3; 4 Lectures]
- Linear transformations: examples, monomorphisms, epimorphisms, isomorphisms, basis gives isomorphism with kn, kernel, image, nullity, rank, cosets of kernel and solutions of f (x) = b, application to linear equations, linear constant coefficient differential equations, matrix of a linear map with respect to bases, change of bases. [Poole 6.4, 6.5, 6.6; 4 Lectures]
- Determinants: properties and methods of calculation (without proofs). [Poole 4.2; 2 Lectures]
- Linear operators: change of coordinates and similarity of matrices, the diagonalization problem, eigenvectors and eigenvalues, characteristic polynomial, diagonalization of n x n matrices with n distinct eigenvalues, application to systems of linear differential equations, Markov chains and Perron-Frobenius Theory. [Poole 6.6, 4.3, 4.4, 4.6; 4 Lectures ]
Textbook
The course is based on the course text which students are strongly advised to buy:
- David Poole, Linear Algebra: a Modern Introduction, Thomson, second edition 2006, International Student Edition: ISBN 0-534-40596-7. (There is a useful website associated with the text).