MAT2103: Linear Algebra II

This course is a foundation course that introduces learners to the basic mathematical concepts. It covers the following topics: Further Linear Transformations; Canonical Forms; Applications of Bilinear Forms; and Inner Product Spaces.

Course Objectives 
On successful completion of this course unit, the learners should be able to: 

• Give the general representation linear transformations; give properties linear transformations; determine kernel and range of linear transformations; as well characterize linear functionals, duals and singularities. 
• Give the elementary canonical forms: characteristic values, annihilating polynomials Characterize; state and prove the Cayley-Hamilton Theorem and work with invariant subspaces 
• Carry out diagonalization: LU, LDLT LDU, PA=LU iv. Demonstrate factorizations, direct sum decomposition, invariant direct sum and primary Decomposition Theorem. 
• Carry out rational and Jordan Canonical forms: cyclic subspaces and decompositions, invariant factors, companion matrices. vi. Demonstrate symmetric and skew symmetric bilinear forms; give their matrix representations; determine ranks and signatures   
• Determine inner products in the Euclidean space; projections; Cauchy Schwartz inequalities; and least squares. 
• Demonstrate Gram-Schmidt orthogonalisation; QR factorization; and applications to systems of differential equations.

Expected Learning Outcome 
This course unit is meant: 

• To discuss the basic competence in the concepts, principles, and procedures of advanced linear algebra and their applications to mathematical analysis and computations. 
• To encourage orderliness, speed and accuracy in the presentation of mathematical expressions in linear algebra. 
• To help learners acquire the skills of expression in proper mathematical language and using mathematical symbols correctly. 
• To provide instruction that contributes to the learners’ abilities to think critically and solve real life problems, to reason mathematically and apply computational skills. 
• To build a strong foundation in mathematical presentation as preparation for subsequent courses in pure and applied mathematics at master’s level.