On successful completion of this subject, students should be able to:

1. Implement Gaussian elimination to solve systems of linear equations.
2. Use and apply vector operations such as the dot product, and concepts such as orthogonality, span, and linear dependence of vectors.
3. Use matrix operations, including matrix multiplication, matrix inversion and LU factorisation.
4. Use and apply concepts related to vector spaces, subspaces and the connection between dimension, linear independence, spanning sets, and basis.
5. Use and apply concepts related to basic linear transformations of the plane, general linear transformations, and the basic notion of kernel.
6. Evaluate determinants using a range of techniques.
7. Be able to compute the eigenvalues and eigenvectors of a linear transformation.
8. Use the concept of orthogonality and orthogonal matrices, including orthogonal projections and the Gram-Schmidt process.

Linear Equations: Introduction to Linear Systems, Gaussian Elimination, applications.
Matrices: matrix operations and rules of arithmetic, elementary matrices, inverses and their calculation, applications.
Determinants: Determinant function, properties, cofactor expansion, Cramer's rule.
Eigenvalues and Eigenvectors: eigenvalues and eigenvectors, diagonalization, applications.
Euclidean Vector Spaces: Vectors in 2 and 3 space, geometric and algebraic interpretation, norms, dot product, cross product, lines and planes. Generalization to Euclidean n-Space, linear transformations between Euclidean vector spaces.
General Vector Spaces: Real vector spaces, subspaces, and concepts of span, linear independence, basis, and dimension.
Inner Product Spaces: Inner product, Cauchy-Schwarz inequality, angle between vectors, orthogonality, Gram-Schmidt Process, least squares, singular value decomposition.
General Linear Transformations: matrix and linear transformations, kernel and range, matrix of linear transformations.