Unit 2
General Vector Spaces

Introduction

In Unit 1, we reviewed the vector space Rⁿ as a generalization of the vector spaces R² and R³. In this unit, we discuss some of the basic properties of general vector spaces. We consider the axioms that define a vector space, examine several examples of vector spaces, define the idea of a subspace, discuss the concept of linear independence, and introduce the ideas of a basis for a vector space and the dimension of a vector space.

The best way to develop a feeling for this material is to work carefully through a number of examples and exercises. Particularly useful examples are those which can be visualized, either directly or by analogy to a case in R² or R³.

Objectives

After completing this unit, you should be able to

1. use the vector space axioms to determine whether a given set of mathematical objects constitutes a vector space.
2. extend a set that does not define a vector space in such a way as to create a set that does do so.
3. determine whether or not a given vector space is a subspace of another given vector space.
4. define the term “linear combination” and determine whether any given vector is a linear combination of members of a given set of other vectors.
5. define the “span” of a set of vectors, and determine when a given subset of vectors “spans” a vector space.
6. define the terms “linear independence” and “linear dependence,” and determine if a given set of vectors is linearly independent or linearly dependent.
7. define the terms “basis” and “dimension,” and compute a basis for any given vector space.
8. compute the row, column, and null spaces of a matrix, and relate these values to the consistency conditions for solution of a system of linear equations.