Math 104.01 (Linear Algebra)
Spring 1999
Plan for Week 8
In the two weeks remaining before Spring Break, we will complete our study
of Chapter 4, the central unit in the course. Again we quote the text (p.
210, emphasis added):
"The mathematical seeds planted in Chapters 1 and 2 germinate and begin to
blossom in this chapter. The beauty and power of linear algebra will be seen
more clearly when we view Rn as only one of a variety of
vector spaces that arise naturally in applied problems. We will find
that a study of vector spaces is not much different from a study of
Rn itself, because we can use our geometric experience
with R2 and R3 to visualize many concepts.
"Beginning with basic definitions in Section 4.1, the general vector space
framework develops gradually throughout the chapter. A goal of Sections 4.3-4.5
is to demonstrate how closely other vector spaces resemble
Rn . ..."
We will see this week how to identify a set of vectors in any vector
space that can play the role of the standard basis vectors
{e1, e2, ... , en}
in Rn -- in particular, that will enable us to represent
any vector in the space in terms of coordinates. The key features
of a basis for a coordinate system are spanning the entire
vector space and linear independence. (It is not required that basis
vectors be of length 1 or mutually orthogonal -- we will take up those properties
much later when we get to Chapter 6.)
This week's lab will explore applications of matrix multiplication and of
bases in several contexts, and will set the stage for the next four labs.
To see the syllabus for Week 8 in a separate window, click
here.
Notes:
-
Your next homework papers will be turned in on Monday, March 8. Those papers
should include solutions to all problems in the assignment below. The assignment
dates are start dates.
-
In general, no solution will be given full credit unless you have written
an explanation of why you know it is correct. (Exceptions to this
rule are the exercises whose numbers appear in parentheses.)
-
In the instructions for Exercises 1-8 in Section 4.3, ignore the references
to R2 -- these exercises are all about
R3.
Assignments
-
Monday, Mar. 1, Sec. 4.3 / #1, 5, 8, 9, 11, 15, 18, 19, 21, 22, 23, 24, 25,
26, 27, 33, 34
-
Wednesday, Mar. 3, Sec. 4.4 / #(1), (3), (5), (7), (9), 13, 15, 16, 17, 21,
22, 27, 32, 33, 34
-
Friday, Mar. 5, Sec. 4.8 / #1, 3, 9, 14, 17, 18, (21), 22, 23
David A. Smith
<das@math.duke.edu>
Last modified: February 11, 1999