Date modified: Sat 2024-02-17 . 11:15 AM
Date created: Mon 2024-02-12 . 07:38 PM
# 00 Introductory remarks. What are we studying in linear algebra? A common organizational scheme when one approaches a subject in mathematics is to look at its (1) **objects**; (2) **collection of said objects**; (3) **functions between said collections**; (4) **theorems and results** pertaining to these objects, collections, and functions; and (5) **methods and techniques** to solve problems. For example, here is one way to view calculus: - Objects: Real numbers. - Concepts: Limits, sequences, series, completeness of the real numbers (axiom of real numbers). - Collections: Intervals. - Concepts: Closed or open intervals. Idea of open sets, closed sets, compact sets, bounded sets. However these are not emphasized until a real analysis course. - Functions: Continuous functions, differentiable functions, integrable functions. - Theorems and results: - Intermediate value theorem for continuous functions - Mean value theorem for differentiable functions - Fundamental theorem of calculus, etc. - Methods and techniques: - Linearization - Differentiation to solve optimization problems - Integration to find areas, volumes, etc. So what about linear algebra? - Objects: Vectors. - Concepts: Linear combination, linear independence, etc. - Collections: Vectorspaces, subspaces. - Concepts: Basis sets, dimension, etc. - Functions: Linear transformation. - Concepts: Kernel and range, one-to-one and onto, invertibility, similarity, etc. - Theorems and results: - Rank-nullity theorem - Theorem of fundamental subspaces ("Fredholm alternative") - Replacement lemma - Invertible matrix theorem - etc. - Methods and techniques: - Gaussian elimination - Pivot analysis in echelon forms - Determinants - etc. Of course, this is not an exhaustive list. But it gives you an idea of our approach. We shall look at linear algebra things, collection of these things, function between these things, what can we say about these things (theorems), and how do we DO the thing, to answer questions. To motivate some of these ideas, let us first look at a type of problem: **Solving system of linear equations**.