Objective:
Present a class of matrices that can be used to illustrate all the topics
covered in a beginning linear algebra course. This demo/student project
can be used as a capstone exploration.
Level: Post
Calculus; linear algebra.
Prerequisites:
A
beginning linear algebra course. All the major topics of the course
including vector spaces and eigen concepts should have been covered. Students
should be familiar with the concepts of symmetric and orthogonal matrices,
inverses, reduced row echelon form, null space, eigenvalues, eigenvectors,
and basis for a subspace.
Platform: Any
software package that includes tools for the major linear algebra computations.
We present aspects of this demo/project for MATLAB, Mathematica, and Maple.
Instructor's
Notes: Throughout a linear algebra course students encounter sets of
matrices which are often described in one of the following ways.
Let S be the set of all m by
n matrices of the form _______.
Let S be the set of all m by
n matrices that have the property ________ .
For example;
S = the set of all 3 by 3 diagonal
matrices.
S = the set of all 5 by 4 matrices whose row
reduced echelon
form has at least one zero row.
S = the set of all nonsingular matrices.
S = the set of all n by n symmetric matrices.
In three of the preceding examples the set S contained
matrices of the same size. The set of all nonsingular matrices contains
square matrices of any size. Both types of sets are used for a variety
of purposes which are designed to foster acquaintance, practice, and reinforcement
of abstract and unifying ideas that are the cornerstone of linear algebra.
By the end of the term students have encountered many sets of matrices
and (hopefully) have acquired a reasonable set of skills that encompass
the areas of matrix
algebra, row operations, inverses, vector space notions, spanning sets,
bases, and eigen concepts. Depending upon the type of course they may also
have dealt with the geometric aspects of linear algebra which provide opportunities
for visualization of a number of topics.
As
instructors we want to have students draw together the topics of the course
and see the interrelationships of the ideas that all too often seem to be
compartmentalized as we progress through chapters of a text. One way to
provide such an opportunity for student learning along these lines is to
use a capstone set of exercises. These may be designed to encompass a number
of topics, require that students experiment with a certain set of matrices,
and quite possibly report findings by writing about properties they discover.
In this regard the use of technology, namely software packages or even
calculators, can be used to provide a format for experimentation. With
such goals in mind an instructor needs a set of matrices that were not
explored in depth previously in the course and which are fairly simple
to generate and manipulate. This demo presents one such set that has
been used successfully for several years with several different text books.
Definition:
A
reversal
matrix is a matrix obtained by writing the rows and columns of an identity
matrix in reverse order.
Example
1.
The 4
by 4 reversal matrix is
.
_______________________________________________________________
Reversal
matrices are simple to construct in a variety of software platforms.
In
MATLAB , to construct the matrix of Example 1 the following single line
of code can be used.
J=zeros(4);
for k=1:4; J(k, 5 – k)=1; end; J
For
an n by n reversal matrix we can use the following commands:
n
= __;J=zeros(n); for k=1:n; J(k, (n+1) – k)=1; end; J
where the value of n is entered in place of the underline.
An m-file, rever.m, can be downloaded for further
convenience.
In Mathematica, to
construct the matrix of Example 1 the following code can be used.
n = 4
J4 = Table[0,{n},{n}]
For[i=1,i<5,J4[[i,(n+1)-i]]=1;i++]
MatrixForm[J4]
To vary n, just change the
value in the first line.
In Maple, the following
commands can be used to construct J4.
with(linalg):
J:=matrix(4,4,0);
for i from 1 to 4 do;J[i,5-i]:=1;od;
eval(J);
For
an n by n reversal matrix we can use the following commands:
with(linalg):
n:=___;#<== enter the size of the reversal matrix
J:=matrix(n,n,0);
for i from 1 to n do; J[i,(n+1)-i]:=1;od;
eval(J);
______________________________________________________________
The following is a sample of the type of capstone
exercise set that can be used with the set of reversal matrices. These
exercises are designed to be used with MATLAB, where command eye
generates an identity matrix, command rref computes the reduced
row echelon form, and command eig can be used to compute eigenvalues
and eigenvectors. The n by n reversal matrix will be denoted as Jn.
You can of course construct a number of other interesting investigations
to suit your students and your course.
(The sample exercises below are just pictures; click
here for a word file with these exercises
and click here for a pdf file of these
exercises.)
For a Maple worksheet involving these exercises click
on reversal.mws . For a Mathematica notebook
click on reversal.nb .
An Instructional Suggestion: It may be helpful
to have a class discussion about reversal matrices before assigning a set
of capstone exercises. The following items could form the basis of such
a discussion.
-
Give the definition of a reversal matrix.
-
Have students explicitly list Jn
for n = 2, 3, 4, and 5.
-
Let S be
the set of all reversal matrices. Ask for a description of this set; for
example the number of members, is S a subspace of some vector space, or
for column vector x with n entries describe the product Jnx.
-
Prove that
the set W = all matrices of the form kJ4
, where k is a real scalar, is a subspace of R4. What is the
dimension of W? Find a basis for W.
-
Provide an
argument that every reversal matrix is diagonalizable.
Laying a foundation for the set of reversal matrices
provides student with a comfort level for the topic. Of course you may
have another set of introductory items that are more in tune for your course.
Credits:
This demo/capstone project was submitted by
David
Zitarelli
Department of Mathematics
Temple University
and is included in Demos
with Positive Impact with his permission. The MATLAB m-file
rever.m was written by David R. Hill. The Mathematica notebook and Maple
worksheet were constructed by Lila F. Roberts.
