LESSON 7: Vector Spaces, Subspaces and Linear Transformations

Terminology:

vector space, subspace, subspace spanned by a set, null space, column space, kernel and range of a linear transformation.

Objectives:

understand the concept of an abstract vector space; learn how to determine if a subset of vector space is a subspace; learn how polynomials are vectors; learn how to determine the null and row spaces for a matrix.

Reading Assignment:

Chapter 4, Sections 4.1-4.2 (pages 209-228).

Lesson Outline

Key Ideas and Discussion:

You might find this lesson difficult because of the theoretical nature of the material. An important thing to remember is that the vectors, as columns-of-numbers, and their properties, that you are already familiar with, are an important example of a general concept: the (abstract) vector space. In this lesson and the next two lessons you will study some important properties of vector spaces, mostly illustrated with examples from the familiar vector space . A second important example of a vector space is , the set of polynomials of degree n.

The properties for an abstract vector space are the same as the properties satisfied by columns-of-numbers vectors. A subspace of a vector space is a special type of subset of the vector space. This special set is characterized by the property that it is closed under addition and scalar multiplication. A consequence of this characterization is that there must always be a zero vector in a subspace. This fact sometimes provides an easy test to determine if a given subset is a not a subspace. If the zero vector is not in the subset, then the subset cannot be a subspace. If the zero vector is in the subset, then further analysis is needed to show whether the subset is a subspace. There is one very important type of subset that is always a subspace: if W is subset of vectors from some vector space V, then the set of all linear combinations of the vectors in W, Span(W), is always a subspace of V. A good way to explicitly describe a subspace is to give a spanning set for the subspace.

Two important subspaces associated with an matrix A are the null space Nul(A), and the column space Col(A). It is easy to see that the column space is a subspace of because Col(A) is just the span of the columns of A. Col(A) is explicitly described as Span(W), where W is the set containing the columns of A. It is more difficult to think of Nul(A) as a subspace because it is defined implicitly as the set of all in that are transformed to by A. But a spanning set for Nul(A) is just the set of vectors that are used in the parametric form for the solutions to , so all you need to do to find an explicit description for Nul(A) is to completely reduce A, and determine the vectors that you would use to describe parametric solutions to . Remember that Nul(A) is a subspace of , but Col(A) is a subspace of . These two subspaces are generalized when T is a linear transformation from a vector space V into a vector space W. In this case, what corresponds to the column space for a matrix is called the range of T and what corresponds to the null space for a matrix is called the kernel of T. Finding the kernel of a linear transformation can sometimes appear difficult, but remember: you can represent linear transformations with matrices. Problems involving non-matrix linear transformations are often solved using matrices.

Practice Problems:

4.1.3, 5, 11, 13, 15, 23 (pp. 217-219); 4.2.1, 5, 9, 13, 15, 25, 31 (pp. 228-230).

Assignment