MATH09003 2017 Applied Linear Algebra

Solve systems of linear equations and analyse the solutions.

Analyse affine transformations in three dimensions.

Interpret the linear algebra in state of the art research publications and reproduce findings.

Explain the use of vector spaces in analysing solutions to systems of equations.

Use projections to find the least squares solution of overdetermined systems.

Decompose matrices into their singular value decompositions and interpret.

Apply matrices to the Fourier transform, graphs and networks.

A lecture will be provided each week. In advance of the lecture, the learner will be asked to review material that is relevant to the lecture so that they get the maximum learning from that lecture.

The project work will challenge the learner to master concepts beyond those covered in the theory lecture.

A terminal exam and continuous assessment will be used to assess the module.

To reinforce the theoretical principles covered in lectures, learners will participate in project work.

The learner will complete a final exam at the end of the semester.

The learner is required to pass both the projects and terminal examination element of this module.

Repeat Exams will be set for Autumn of each year.

Repeat project work can be submitted at the repeat exam sitting.

Review solutions of linear equations using Gauss Elimination / Back-substitution / Elementary Row operations / pivots / REF / Gauss-Jordan elimination / RREF

- Consistency and systems with unique/multiple/zero solutions

- (Geometry of Solutions) (solutions of equations as points, lines and planes)

- Complete solution to AX = B (column space containing b, rank and nullspace of A and special solutions to AX = 0 from row reduced R)

Matrix Algebra

Review addition and multiplication of matrices and vectors and their transposes. 

- Inverse matrices. Calculation via Gauss Elimination. Invertibility. 

- Block multiplication

3D Vectors/Rotation

- Inner product, length, orthogonality cosine rule/angles, cross product

- Rotation matrices, Euler angles and non-commutativity

- Projection, reflection, scaling

- Lines / planes in 3 dimensions

- Linear transformations and change of basis (dual basis)

Vector spaces and subspaces

- Basis and dimension / linear independence and span

- Four fundamental subspaces ( row, column, right-left nullspaces, and their bases)

Least squares solutions, overdetermined systems (closest line by understanding projections)

Orthogonalization by Gram-Schmidt (factorization into A = QR)

Orthonormal matrices  (rotation matrices as)

Determinants / as volume / cofactor formula / applications to inverses / calculation via Gauss elimination

Eigenvalues / SVD

- Eigenvalues and eigenvectors (diagonalizing and computing powers of matrices)

- Matrix exponentials. Solving difference and differential equations

- Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for x'Ax > 0, applications) A=QLQT

- Singular Value Decomposition (SVD)

- The Moore-Penrose Pseudoinverse. Least solution underdetermined system.

LU Factorisation

Skew-symmetric matrices / skew matrices

Fourier matrices / fast Fourier transform

Graphs and networks

Kronecker product