Page 162 - FINAL_HANDBOOK_20252026

Faculty of Science Handbook, Academic Session 2025/2026

SIM3001 SIM3004
GRAPH THEORY ADVANCED LINEAR ALGEBRA

Eulerian graphs, trees, planar graphs, graph Inner product spaces, the Cauchy-Schwarz
colouring and chromatic polynomials, inequality, the Gram-Schmidt
Hamiltonian graphs, matching theory, orthogonalization process, orthogonal
directed graphs and the shortest path complements, orthogonal projections.
problem, network theory. Adjoint operators, normal operators, self-
adjoint operators, unitary operators,
Assessment: positive definite operators. Bilinear forms,
Continuous Assessment: 40% congruence, ranks, Sylvester’s law of
Summative Assessment: 60% inertia, classification of symmetric bilinear
forms, real quadratic forms. The Schur
SIM3002 triangularization theorem, the primary
COMBINATORIAL MATHEMATICS decomposition theorem, the Jordan
canonical form.
Enumerative combinatorics: permutations
and combinations, Catalan numbers,
Stirling numbers and partition numbers. Assessment:
Continuous Assessment: 40%
Existential combinatorics: pigeonhole Summative Assessment: 60%
principle, Ramsey theory of graphs and
systems of distinct representatives. SIM3005
MATRIX THEORY
Combinatorial designs: block designs,
balanced incomplete block designs, Steiner Rank and nullity of matrices, Sylvester’s law
triple systems and Hadamard matrices. inequality, the Frobenius inner product, the
Gram-Schmidt process, the continuity
Assessment: argument. Rank and full rank
Continuous Assessment: 40% decompositions, LU and QR
Summative Assessment: 60% decompositions, spectral decompositions,
singular value decompositions, polar
decompositions, Cholesky decompositions.
SIM3003 Generalized inverses, Moore-Penrose
NUMBER THEORY inverses, the best approximation solutions,
least squares solutions. Kronecker products
Prime numbers. The division algorithm and of matrices, permutations, matrix functions
unique factorization theorem for integers. of Kronecker products, Schmidt ranks and
Linear diophantine equations. Theory of decompositions.
congruence and the Chinese Remainder
Theorem. RSA encryption. Quadratic Assessment:
reciprocity and the Legendre symbol. Continuous Assessment: 40%
Arithmetic functions. Primitive roots. Summative Assessment: 60%

Assessment:
Continuous Assessment: 40%
Summative Assessment: 60%

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