Unit 9
General Vector Spaces

In Unit 8, we considered Euclidean n-space. In this unit, we generalize to arbitrary vector spaces. We do so by defining a vector space as any set of objects for which operations of addition and multiplication by a scalar are defined, and which satisfies a set of ten axioms modeled on the properties of Euclidean n-space. Following this discussion, we introduce the concept of a subspace, together with the definition of the “span” of a set of vectors.

Another useful concept in the study of vector spaces is “linear independence.” As the term suggests, two vectors are linearly independent if they do not define the same line. More generally, a set of vectors is linearly independent if the vectors do not lie in a linear subspace with fewer dimensions than the number of vectors. It turns out that a basis for a vector space is both the smallest set of vectors which span the space, and the largest set of linearly independent vectors that can be contained in the space. In addition, knowing that a particular set of vectors is linearly independent is a powerful tool in proving theorems about that set of vectors.

The final topic for this unit is the definition of the row, column, and null spaces for a matrix, and a discussion of the remarkable way in which these vector subspaces relate to systems of linear equations.

Objectives

When you have completed this unit, you should be able to

1. use the vector space axioms to determine whether or not any given set of objects constitutes a vector space.
2. give the definition of a “subspace of a vector space,” and determine whether or not a given subset of a vector space is also a subspace.
3. define the “span” of a set of vectors.
4. define the terms “linear combination of vectors,” “linear dependence” and “linear independence,” and determine which of the last two terms applies to any given set of vectors.
5. define what it means for a set of vectors to be a “basis” for a vector space, and determine whether or not any given set of vectors in a vector space forms a basis.
6. define the “dimension” of a vector space and prove that any set of vectors containing more elements than the dimension of the vector space in which they are contained is necessarily linearly dependent.
7. define the “row space,” “column space” and “nullspace” of a matrix, and determine bases for these spaces.
8. define the “rank” of a matrix and compute the rank of any given matrix.
9. use information about the column space of a matrix A to determine whether or not the system of equations Ax = b is consistent.