Authorised by Academic Registrar, April 1996
Objectives The objectives of the subject GAS2613 are for students to understand the basic properties of vector spaces; determine whether a given subset of a vector space is also a vector space; determine the matrix representation of a linear transformation on a finite dimensional vector space; find the inner product of two vectors in a finite dimensional vector space; find the inner product of two functions, considered as vectors, in an infinite dimensional vector space; find the orthogonal projection of a vector in an n-dimensional vector space onto a subspace; find the mean-square approximation of a continuous function by an n-th degree polynomial; use the Gram-Schmidt process to construct an orthonormal basis; find the eigenvalues and eigenvectors of square matrices; diagonalise symmetric matrices associated with quadratic forms, and apply it to the study of quadric surfaces; use the power method to approximate the dominant eigenvector of a square matrix, determine the corresponding eigenvalue, using Rayleigh's quotient; reduce a symmetric matrix to a tri-diagonal form, using Householder transformations, and find its eigenvalues.
Synopsis This subject aims to continue the study of linear algebra beyond subject GAS1612, emphasising the general concepts of a vector space and the particular case of an inner product space as unifying threads in mathematics. Linear spaces: general concepts, basis and dimension, linear transformations, inner product spaces; orthogonalisation and projection; matrix algebra: diagonalisation theorems for real symmetric matrices, quadratic forms, applications to analytical geometry, numerical methods of eigenvalue analysis for real symmetric matrices.
Assessment Assignments: 40% + Examination: 60%