**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, skew-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, volume element, Hodge star operator.

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

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

Jun 10, 1998