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| Module Availability |
Autumn / Spring |
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| Assessment Pattern |
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Unit(s) of Assessment
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Weighting Towards Module Mark( %)
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Coursework: 2-3 tests in Autumn semester
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50%
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Coursework: 1 test in Spring semester
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10%
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Exam: End of Spring semester
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40%
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Qualifying Condition(s)
A weighted aggregate mark of 40% is required to pass the module.
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| Module Overview |
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| Prerequisites/Co-requisites |
| None |
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| Module Aims |
The aim of this module is first to introduce the basic concepts and techniques of vector and matrix calculations, and their applications to geometry and differential equations, and then to use these to motivate the more abstract theory of linear algebra. |
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| Learning Outcomes |
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At the end of the module the student should:
· Know the terminology associated with vectors in two and three dimensions, and be able to find equations of straight lines and planes. Use vector methods to prove simple geometrical results.
· Be able to solve systems of simultaneous linear equations.
· Be able to manipulate matrices and determinants, and find eigenvalues and eigenvectors.
· Be able to solve first order systems of homogeneous linear ODEs.
· Know and understand the terminology associated with vector spaces
· Understand and be able to apply the connection between linear maps and matrices. Appreciate the significance of the eigenvalues and eigenvectors of a linear map.
· Understand the concepts of inner product and orthogonality, for both real and complex vector spaces.
· Recognise the equation of a conic, and find its principal axes.
Formulate simple proofs of results similar to those covered in the course.
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| Module Content |
· Vector algebra; scalar, vector and triple products; equations of straight lines and planes.
· Geometrical proof by vector methods (e.g. proofs of concurrency and collinearity).
· Matrices and simultaneous linear equations. Solution by Gaussian elimination.
· Determinants and invertibility. Eigenvalues and eigenvectors. Matrix diagonalisation. Matrix exponentials and Abel’s formula.
· Linear operators. Homogeneous and inhomogeneous linear ODEs.
· Systems of first order linear ODEs (homogeneous, constant coefficient). Classification of 2 x 2 systems.
· Introduction to Vector Spaces. Subspaces, linear independence, bases, dimension.
· Linear transformations. Matrix representation. Kernel and image, rank and nullity. Eigenvectors of a linear map.
· Bilinear and quadratic forms. Inner products (real and complex). The Cauchy-Schwartz and Triangle inequalities. Norms, orthogonality, the Gram-Schmidt process.
Geometry of conics, principal axes. |
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| Methods of Teaching/Learning |
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Teaching is by lectures, tutorials and tests. Learning takes place through lectures, tutorials, tests and exercise sheets.
40 hours of lectures and tutorials over 10 weeks in the Autumn semester
40 hours of lectures and tutorials over 12 weeks in the Spring semester.
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| Selected Texts/Journals |
Recommended:
J. Gilbert and C. Jordan (2002), Guide to Mathematical Methods, Palgrave Macmillan, ISBN 0333794443
S. Goode and S. Annin, Differential Equations and Linear Algebra, Pearson, ISBN 0131293397
David Poole, Linear Algebra – A Modern Introduction, Brooks Cole (2002), ISBN 0534341748.
Also useful:
C. McGregor, J. Nimmo and W. Stothers (1994 / 2000), Fundamentals of University Mathematics, Albion Publishing / Horwood, ISBN 1898563101
H. Anton : Elementary Linear Algebra, Wiley (2000), ISBN 0471170550.
R.B.J.T. Allenby, Linear Algebra, Butterworth-Heinemann (1995), ISBN 0340610441 |
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| Last Updated |
13 October 08 |
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