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FALL 2009
Lab 4: Linear Systems
This Maple lab is based in part on earlier versions prepared by Professors R. Falk and R.
Bumby of the Rutgers Mathematics department.
Introduction.
In this lab we use Maple to find
eigenvalues
and
eigenvectors
of matrices, and
solve linear systems of algebraic equations and systems of first order linear differential equations.
We also obtain pictures of the slope fields of these equations in the
phase plane
.
Please obtain the seed file from the web page and save it in your directory on eden.
Then
prepare the Maple lab according to the instructions and hints in the introduction to Lab 0. Turn in
only the printout of your Maple worksheet, after removing any extraneous material and any errors
you have made. For Lab4 you will also need to obtain a
supplementary worksheet
that contains
some commands which you are asked to try out but which should not appear in the final lab.
0. Setup.
As usual, the seed file begins with commands which load the required Maple packages:
with(plots):
and
with(DEtools):
. In this lab we also require a Linear Algebra package, which
is loaded using the command
with(LinearAlgebra):
.
1. Matrix entry.
The seed file and the supplementary worksheet include the definitions of three
matrices,
A
,
B
and
C
:
A:= <<3212>,<4914>>;
B:= <<3,4,6><2,3,4>>;
C:= <<10,0><2,6><7,6>>;
The syntax for entering matrices into Maple should be clear from these examples (and the resulting
matrices); if not, you can read about it by entering
?<
at the prompt. For practice, enter in your
worksheet (i) a column vector with three entries, all different; (ii) a row vector with four entries,
all different; and (iii) a 3
×
3 matrix whose entries are all distinct.
2. Matrix Operations.
The goal of this section is to understand how the matrix operations of
addition (+), matrix multiplication (.), scalar multiplication (
*
), and powers ( ˆ ) act in various
circumstances. We will experiment with these, and since some of our experiments may give errors
and unexpected results, we will do so in the
supplementary worksheet
.
The worksheet that you
submit should contain only a
discussion
, guided by the questions below.
A few examples of the use of matrix operations are already in the supplementary worksheet,
and you should add others to allow a full discussion of these operations. Some of these examples
will lead to errors, and such errors will find their place in the worksheet discussion: your comment
should include a description of the failed command and your interpretation of the error message.
If you have any doubt about your interpretation of a result, you can consult
Maple help
. There
are various ways to do so (see the
Help
button at the top of the worksheet) but here are two
particular ones that may be helpful: at a command prompt
>
, typing “?anything” (
> ?anything
)
will give help on the topic “anything” (try
> ?LinearAlgebra
or
> ?.
), and placing the cursor on
a command you have typed, and pressing F2, will give help on that command.
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 Fall '09
 Ming
 Linear Algebra, Matrices, Matrix Operations, matrix exponential function

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