To introduce the student to the Laplace Transform, Fourier Series, and their use in solving Ordinary Differential Equations. To introduce the student to the theory and methods of Linear Algebra. To give the student a broad understanding of the numerical processes used in solving Linear Algebra problems, and their extension to some nonlinear problems

Laplace Transforms, Transform Theorems, Convolution, the Inverse Transform. Fourier Series functions of arbitrary period, even and odd functions, half-range expansions. Application of Laplace transforms and Fourier series to finding solutions of ordinary differential equations. Vector Spaces, linear independence, spanning, bases, row and column spaces, rank. Inner Products, norms, orthogonality. Projection theorems and applications, e.g. least squares, and fitting data with orthogonal polynomials. Eigenvalues and eigenvectors. Diagonalisability. Symmetric matrices, including numerical methods to diagonalise same. Numerical solution of systems of linear equations : Gauss elimination, LU-decomposition, Cholesky decomposition, pivoting, iterative improvement, condition number; iterative methods including Jacobi, Gauss-Seidel and S.O.R.

On successful completion of this module students will (will be able to): 1. Use Laplace transform method and tables to solve linear ordinary differential equations and convolution type integral equations. This will be assessed by midterm and final written exam. 2. Obtain Fourier series expansions of periodic functions and functions defined on finite domains. This will be assessed by midterm and final written exam. 3. Determine whether sets of vectors are linearly independent, span a space, or form a basis. This will be assessed by final written exam. 4. Given a vector space and a basis for a finite dimensional subspace, produce an orthonormal basis and find best approximation to an element of the vector space in this subspace.This will be assessed by final written exam. 5. Find eigenvalues and eigenvectors of a given matrix, and diagonalise a matrix where possible. This will be assessed by final written exam. 6. Use factorisation and iterative methods to solve systems of linear equations and determine when iterative methods will converge. This will be assessed by final written exam.

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