# MATH09003 2017 Applied Linear Algebra

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**Description**

The subject covers the linear algebra required for post-graduate engineering courses. The learner will gain the expertise to interpret the linear algebra models used in the engineering literature. It will also enable learners to model problems using linear algebra methods.

### Learning Outcomes

*On completion of this module the learner will/should be able to;*

**1.**

*Solve systems of linear equations and analyse the solutions.*

**2.**

*Analyse affine transformations in three dimensions.*

**3.**

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

**4.**

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

**5.**

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

**6.**

Decompose matrices into their singular value decompositions and interpret.

**7.**

Apply matrices to the Fourier transform, graphs and networks.

### Teaching and Learning Strategies

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.

### Module Assessment Strategies

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 Assessments

Repeat Exams will be set for Autumn of each year.

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

### Indicative Syllabus

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

### Coursework & Assessment Breakdown

**Coursework & Continuous Assessment**

**End of Semester / Year Formal Exam**

### Coursework Assessment

Title | Type | Form | Percent | Week | Learning Outcomes Assessed | |
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1 | CA 1 | Coursework Assessment | Assessment | 30 % | Week 6 | 1,2,4 |

2 | Project | Coursework Assessment | Project | 30 % | Week 12 | 1,2,3,4,5,6,7 |

### End of Semester / Year Assessment

Title | Type | Form | Percent | Week | Learning Outcomes Assessed | |
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1 | Terminal Exam | Final Exam | Closed Book Exam | 40 % | End of Semester | 1,2,3,4,5,6,7 |

### Full Time Mode Workload

Type | Location | Description | Hours | Frequency | Avg Workload |
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Lecture | Lecture Theatre | Lecture | 2 | Weekly | 2.00 |

Practical / Laboratory | Computer Laboratory | Laboratory Practical | 2 | Fortnightly | 1.00 |

Independent Learning | Not Specified | Independent Learning | 7 | Weekly | 7.00 |

### Online Learning Mode Workload

Type | Location | Description | Hours | Frequency | Avg Workload |
---|---|---|---|---|---|

Lecture | Online | Theory Lecture | 1 | Weekly | 1.00 |

Independent Learning | Not Specified | Independent Learning | 8.5 | Weekly | 8.50 |

Practical / Laboratory | Online | Online lab | 0.5 | Weekly | 0.50 |

### Required & Recommended Book List

**Recommended Reading**

2013-09-03

*Coding the Matrix: Linear Algebra through Applications to Computer Science*Newtonian Press

ISBN 0615880991 ISBN-13 9780615880990

**Recommended Reading**

2007-11-01

*Computational Science and Engineering*Wellesley-Cambridge Press

ISBN 0961408812 ISBN-13 9780961408817

Computational Science and Engineering This book encompasses the full range of computational science and engineering from modelling to solution, whether analytic or numerical. Full description

**Recommended Reading**

2016-08-11

*Introduction to Linear Algebra*Wellesley-Cambridge Press

ISBN 0980232775 ISBN-13 9780980232776

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