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, S.O.R., and Conjugate Gradient methods. Extension to nonlinear systems using Newton's method.