|Credits: 10||Convenor: Dr. N. J. Snashall||Semester: 1 (weeks 7 to 12)|
|Assessment:||Individual and group coursework: 20%||One and a half hour exam: 80%|
To understand, reconstruct and apply the main results and proofs covered in the module.
To decide whether a vector space has a basis of eigenvectors for a given linear transformation.
To choose a basis with respect to which the matrix of a linear transformation has a particularly manageable form.
To construct an orthonormal basis for a given finite-dimensional inner product space.
To recognise Hermitian, unitary and normal matrices.
To work in a group context.
The application of mathematical principles and concepts to new situations.
Written presentation of mathematical arguments in a coherent and logical form.
Use of techniques from the module to solve problems.
Experience of working as part of a team.
Linear transformations, and the kernel and image of a linear transformation. Rank and nullity and their relationship. The dual of a vector space.
The matrix of a linear transformation. Change of basis matrix. Eigenvalues, eigenvectors and eigenspaces.
Characteristic polynomial (including some discussion of the fundamental theorem of algebra and working over the complex field). The Cayley-Hamilton theorem.
The minimum polynomial and its relationship to the characteristic polynomial. Relationship between the minimum polynomial and the existence of a basis of eigenvectors and diagonalizability of matrices.
Inner product spaces (real and complex), Gram-Schmidt Process. Hermitian, unitary and normal transformations (with discussion of the existence of an orthonormal basis of eigenvectors for such transformations, and application to quadratic forms).
R .B. J. T. Allenby, Linear Algebra, Edward Arnold.
C.W. Curtis, Linear Algebra, an Introductory Approach, Springer.
J .B. Fraleigh and R. B. Beauregard, Linear Algebra, 3rd Ed., Addison-Wesley.
S. Andrilli and D. Hecker, Elementary Linear Algebra, 2nd Ed., PWS-Kent.
W. K. Nicholson, Linear Algebra, with Applications, 3rd Ed., PWS-Kent.
There are four questions on the examination paper; all marks gained will be counted. All questions carry equal weight.