**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**-

Thu May 28 14:15:25 PDT 1998