Download Lab: Linear Systems

Lab: Linear Systems
Easy version for a classroom situation: Set up exact polynomial fit. This is an excellent
reinforcement of the link between the geometric view of a polynomial function and the
algebraic view of a polynomial function.
To start: In DERIVE or another graphing utility, show a parabola together with three
points on the curve labeled with their coordinates. The parabola should be labeled
generically, as y  ax 2  bx  c. Work with the students to identify the desired unknowns
in their polynomial equation, and what information they do have to extract linear
equations in the unknowns a, b, and c. Prior to using a computer algebra system such as
DERIVE to find the solution set, ask the students what requirements on the points are
necessary to insure the existence of a unique solution. Then solve the system, and
compare the resulting polynomial to that which you have graphed and also to DERIVE’s
FIT algorithm, and also a calculator model if a student is familiar with calculator
regression. Point out the distinction between exact fit of n + 1 data points for a
polynomial of degree n and the approximate fit used for more data.
Screenshot of a demo on exact polynomial fit
Homework: Use the students’ phone numbers to create “randomized” data and have them
fit polynomials to the points. Then move on to other applications of linear systems, such
as the following:
Linear Systems Application: Traffic Flow
SOURCE: Grossman, Linear Algebra, 5ed, Brooks/Cole, 1994.
As a civil engineer, suppose you have measured traffic flowing on various streets. You
know how this is done; someone puts out a hose that you drive over. There are some
assumptions you make when this data is collected:
a. The time during which the data is collected is uniform for every street measured.
b. The traffic that flows out of an intersection equals the traffic flowing into the
intersection (no accidents or miracles of flight).
c. The streets are all one-way (a two way street is soluble, but will increase the
dimension of the matrix involved in the problem).
Now suppose you have gathered the following data. It is summarized in the directed
network graph presented below. The numbers and variables represent the number of cars
passing a sensor each hour. Notice that you don’t have a complete set of data; but a linear
system of equations will help you solve this problem:
1. Set up a linear equation for each intersection, based on the assumption (b) above.
For example, at intersection [1] in the diagram, we have:
traffic in = x1  x2 = 70 + 50 = traffic out, or the linear equation
x1  x2  120.
2. Set up a matrix representing the augmented matrix of your linear system. In
DERIVE, you do this by clicking on the matrix icon, specifying dimension, then
typing in entries (TAB to navigate between entry areas). To row reduce your
matrix, type ROW_REDUCE(#1) (use the line number of your augmented
matrix) and Simplify.
3. The mathematician is satisfied with modeling the problem and using DERIVE as
the mathematical assistant. The engineer needs to interpret the solution. Here are
some issues to consider: (i) If there are infinite solutions, do all of them make
sense? That is, can a value xi be negative in this situation? You should be able to
come up with a bound on the number of cars for each of the streets. (ii) What
would the engineer make of a situation where there are NO solutions? (iii)
Suppose you need to close a section of a street, for example, set x1  0. How
could you modify your linear system to determine what the implications would
be? (There are clever ways to eliminate a column from a matrix in DERIVE, but
you can just create a new matrix too.)
Directed graph representation of traffic flow