MAT2103 Linear Algebra II

This course is a foundation course that introduces learners to the basic mathematical concepts such as Further Linear Transformations, Canonical Forms, Applications of Bilinear Forms, and Inner Product Spaces among others.

General 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
• 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.
• 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 Outcomes
At the end of this course unit, students will be able:

• 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.