Linear Algebra and Applications

Overview

Linear algebra and applications is the second of two mathematics core units and builds upon and expands the mathematical concepts developed in Calculus and Applications. It will provide a grounding and further understanding of mathematics and mathematical processes essential in underpinning the skill and knowledge necessary to carry out more complex computations.

Unit learning outcomes

Students who successfully complete this unit will be able to:

1. Perform simple operations involving determinants, the rank of a matrix and its null space (K2)
2. Use the methods of Gaussian elimination, Cramer’s rule and inverse matrices to solve systems of linear equations and apply them to relevant examples (K2, S1)
3. Perform operations with vectors and have a working understanding of vector spaces. Use vectors to calculate scalar and vector products, determine linear (in)dependence of vectors (K2, S1). Use matrix linear transformations.
4. Describe straight lines and planes in three dimensions and the relationships between them (K2, S1)
5. Use curvilinear coordinates, linear transformations and conversions with parametric forms to solve simple problems. Determine the curvature and radius of curvature for a curve, angular velocity and torque (K2,S1)
6. Use complex numbers to solve equations, describe graphically complex numbers in the Argand plane. (K2)

Teaching methods

Hawthorn

Assessment

Content

• Matrices: basic matrix algebra, multiplication of matrices, special matrices, determinants, inversion, rank, null space, basis, linear independence.
• Linear transformations: matrix of linear transformations, examples: rotations, inversions and projections. Applications to fundamental examples 
• Matrices: Systems of linear equations: elementary row operations, augmented matrix, row echelon form, Gaussian elimination, Cramer’s rule, inversion method, solution in the parametric form. Examples and applications to relevant models. 
• Elements of linear geometry: equation of a line, equation of a plane, intersection between lines and planes, distance from a point to a plane, distance from a point to a line, distance between two lines. 
• Elements of differential geometry: curvilinear coordinates, curves and their properties, curvature, velocity and acceleration. Quadric Surfaces. The stationary points of a function of two variables 
• Complex numbers and their properties: imaginary numbers, complex conjugates, Argand plane in Cartesian and polar forms, de Moivre’s theorem, roots of complex numbers, complex exponential form and applications.