Aims & Learning Objectives:
Aims: To teach the definitions and basic theory of abstract linear algebra and, through exercises, to show its applicability.
Students should know, by heart, the main results in linear algebra and should be capable of independent detailed calculations with matrices which are involved in applications. Students should know how to execute the Gram-Schmidt process.
Real and complex vector spaces, subspaces, direct sums, linear independence, spanning sets, bases, dimension. The technical lemmas concerning linearly independent sequences.
Dimension. Complementary subspaces. Projections.
Linear transformations. Rank and nullity. The Dimension Theorem.
Matrix representation, transition matrices, similar matrices. Examples.
Inner products, induced norm, Cauchy-Schwarz inequality, triangle inequality, parallelogram law, orthogonality, Gram-Schmidt process.