This module has two aims: 1. to introduce basic matrix techniques for the formulation and solution of engineering problems. These techniques include the representation of geometric transformations, the power method to find eigenvalues and the solution of systems of differential equations. 2. to use partial differentiation to formulate and solve optimisation problems in several variables and to perform error analysis.
Linear transformations: Representation by matrices. Dilation, projection, rotation and translation. Parametrized transformations. Inverse of a matrix. Eigenvalues and eigenvectors and their geometric interpretation.
Linear Algebra: Calculation of maximum and minimum eigenvalue using the Power Method for 2D and 3D matrices.
Systems of first order ODE’s: Review of first order and one variable ODE’s. Coupled first order DE’s – application to heat transfer. Conversion of higher order DE’s to systems of first order ones. Implementation of Euler Method in Matlab for solving systems of such equations
Coupled second order DE’s without friction terms: Application to coupled spring systems. Application of matrix techniques to find normal modes and frequencies.
Applications:
Functions of more than one variable: Geometrical representation. Partial Differentiation. Higher derivatives. Optimisation for functions of several variables, Calculation of experimental error.
