MTH 525, FALL 2006

Brief Notes for Weeks 1 & 2

 

Brief outline for Week 1:

Monday, Aug 28

           We cover most of chapter 1.  (Sections 1.2 through 1.6 should be review.  Please read sections 1.2-1.6 before coming to class.)

           I discuss fields, why we need them, how to construct them.

           I review elementary row operations (EROs) and the three types of EROs.

           We briefly discuss matrix multiplication, viewing ÒpremultiplicationÓ as Òrow manipulationÓ.

           In preparation for WednesdayÕs class, please do the following problems:

           Do: 5: 2, 5, 10: 2, 3, 5, 15: 1, 8, 21: 3, 4, 26: 1, 5, 8.

 

Wednesday, August 30

           We discuss properties of invertible matrices (section 1.6.)

           We begin chapter two, discussing abstract vector spaces, focusing on examples.  Subspaces are mentioned.  (Sections 2.1-2.2.)

 

           The major concepts from Chapter 1 are:

                       Definitions and examples of fields.

                       Linear combinations and ÒequivalenceÓ of systems of linear equations.

                       Row equivalence of matrices and, ultimately, the unique representative of this class, the row-reduced echelon matrix.

                       Matrix products and invertible matrices.

 

Week 2

Wednesday, Sept 5

           Assignment 1 is collected at the beginning of class.

           We continue discussion on vector spaces and bases, including coordinates of a vector with respect to a fixed basis.  (Sections 2.2, 2.3, 2.4)

           Important definitions are those of vector space, span, linearly independent, basis, dimension.

           We cover theorems 1, 2, and 3.  These theorems give a short way to identify subspace, show that subspaces are closed under intersection and that the span of a set is a subspace.

           Then we prepare for the concept of dimension by proving that if a vector space is spanned by n elements, no independent set can be larger (Theorem 4), leading to the corollary that all bases (of finite subspaces) are the same size. 

           A nice lemma (before Theorem 5), page 45, shows that any independent set not spanning the vector space, can be extended to a larger independent set.  As a corollary (Theorem 5), we see that every independent set can be extended to a basis.

           We prove that the set of coordinates of a vector with respect to an ordered basis is unique.

 

           In preparation for next MondayÕs class, please do the following problems:

           Do: 33: 3, 6, 7, 39: 1, 2, 3, 5, 8 48: 2, 3, 6, 54: 1-5.

 

Last modified October 1, 2006