**Duration:**

**Number of lectures per week:** 3

**Assessment:** Some assignments, which do not contribute to the final grade

**End-of-year Examination:** One 3-hour end of year examination

**Description: **
Vector space over a field, coordinate functions
with respect to a basis, dual space, M and M^{*} mutually dual,
homogeneous linear equations, linear operators, matrix with respect to a basis,
Einstein index notation, conjugacy classes, trace, characteristic
polynomial, rank, direct sums, invariant subspaces,
eigenspaces, diagonalisability, simultaneous diagonalising,
Hamilton-Cayley, primary decomposition theorem, Jordan form,
minimum polynomial, zeros
of minimum polynomial, diagonalizability iff minimum polynomial is a product
of distinct linear factors.

Bilinear forms, symmetric, hermitian,
non-singular scalar product space, covariant and contravariant components,
raising and lowering indices, M = NÅN^{^} if N
finite dim non-singular, classification of scalar products: complex
symmetric, real symmetric, hermitian, diagonalisation, quadratic forms,
Sylvester law, diagonalisation using determinants (Jacobi), condition
for positive definite, Schwarz and triangle unequalities, adjoint operator,
self-adjoint, isometry, normal operator, spectral theorem, orthogonal,
unitary,
Lorentz transformations, simultaneous reduction of quadratic forms.

Tensor addition, multiplication, contraction, construction of invariants, wedge product of skew-symmetric tensors, categories and functorial properties, volume element, Hodge star operator.

Tensor fields, metric tensor and line elements, gradient vector, tangent and normal, orientation, volume and area elements.

Exterior derivative, closed and exact forms, integration of forms.

n-dim vector analysis with wedge product and Hodge star operator, Poincare lemma, Gaussian curvature, Stokes's theorem and applications.

Mar 27, 2003

File translated from T

On 27 Mar 2003, 10:28.