The aim of this module is to introduce some more advanced concepts in Linear Algebra and Numerical Linear Algebra

Complex vector Spaces: Review of real vector and innner product spaces. Complex inner product spaces. Gram-Schmidt process. Unitary, normal and Hermitian matrices. Eigenvectors and eigenvalues. Diagonalisability. Schur's Lemma. Jordan Canonical form. Singular value decomposition. Introduction to Function spaces. Normed spaces and Banach spaces. Standard examples such as C([a,b]) and sequence spaces. Bounded linear operators and continuous linear functionals. Operator norms. Hilbert space and Riesz representation theorem. Numerical Linear algebra. Krylov subspace methods. Foundations of Conjugate Gradient method. Other iterative methods for solutions of systems of equations. Application of Krylov subspace methods to finding eigenvalues. Lanczos algorithm. QR factorization.

The student will be able to carry out calculations involving inner products in real and complex vector spaces. The student will understand the importance of an orthonormal basis and be able to convert a basis to an orthonormal basis by means of the Gram-Schmidt orthogonalisation process. The student will be able to identify types of matrices, such as symmetric, unitary, normal and Hermitian and know the properties of their eigenvalues. The student will know how matrices can be represented in different forms, e.g. Jordan form, singular value decomposition, diagonalisable. The student will understand the idea of a norm on a vector space V in the cases where V is R^n or C^n or C([a,b]) or l ^p. The student will understand the idea of completeness. The student will be able to calculate operator norms for linear operators between finite dimensional vector spaces. The student will understand the concept of the dual of a normed vector space and do calculations to determine the ralationship between the space and its dual. The student will be able to do calculations using the Conjugate Gradient method to solve systems of equations and understand how different norms can be used in a profitable way. The student will be able to carry out other iterations to solve systems of linear equations and to determine eigenvalues.

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