School of Mathematics
Course 211 - Linear algebra & Differential Forms 1998-99 (SF Mathematics, SF Theoretical Physics, SF Two-subject Moderatorship with Economics & JS Two-subject Moderatorship )
Lecturer: Prof. D.J. Simms
Requirements/prerequisites: 111, 131

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 = NN^ 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