motivating students for linear algebra by using...
TRANSCRIPT
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Motivating Students for Linear Algebra by using Puzzles
Dr. Naima Naheed
Benedict College
Columbia, SC
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Introduction
Puzzle 1: ADDITION TABLE
• Purpose: To learn rank, free variables, and consistency
Puzzle 2: GOSSIP NETWORK
• Purpose: To understand the application of matrix multiplication
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Given a, b, c, and d, find all possible values of r,s,t and u to fill in the
following addition table
+ t u
r
s
a c
b d
We have r + t = a
s + t = b
r + u = c
s + u = d
(1)
(2)
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Writing equations (2) for the puzzle in (1) in matrix form gives
The echelon form of matrix A is
d
c
b
a
u
t
s
r
1010
1001
0110
0101
acbd
ca
b
a
u
t
s
r
0000
1100
0110
0101
A
Rank=# of
nonzero
rwows=3
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The system becomes consistent
0acbd cbda
Number of free variables=number of unknowns - rank(A)=4-3=1
Infinite number of solutions
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Example : 2X2 Puzzle
+
r
s
t u
3 5
4 6
+ -2+u
5-u
6-u
3 5
4 6
u
+ -1 1
4
5
3 5
4 6
Let u=1
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Example: 3X3 Puzzle
+
r
s
t
u v w
6 3 5
5 2 4
7 4 6
+
5-w
4-w
6-w
1+w -2+w w
6 3 5
5 2 4
7 4 6
Let w=1
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+ 2 -1 1
4
3
5
6 3 5
5 2 4
7 4 6
3 X 3 Puzzle
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Generalization of the
+ u n k n o w n
u
n
k
n
o
w
n
G
I
V
E
N
n
n
nn puzzle
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nnn 2
2nnn
unknowns
equations
2111 nnn No. of 2 x 2 blocks
Rank (A)= 12122 nnn
No. of free variables= 1)12(2 nn
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Puzzle 2: Seven people are all connected by e-mail. Whenever one
of them hears a juicy piece of gossip, he or she passes it along by
emailing it to someone else in the group according to following table:
Sender Recipients
Amy Bob, Gal
Bob Clara
Clara Don, Gal
Don Efaz
Efaz Fiin
Fiin Amy
Gal Ehaz
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Amy Bob
Clara
Don Efaz
Fiin Gal
If Amy hears a rumor, how
many paths will it take for
everyone else to hear the
rumor?
Diagraph G
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Adjacency matrix: If G is a diagraph with n vertices, then the
adjacency matrix is defined by
1 if there is a directed edge between vertices i and j
a ij0 otherwise
=
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Letting people Amy through Gal correspond to vertices 1 through 7 in
alphabetical order, the adjacency matrix is
0010000
0000001
0100000
0010000
1001000
0000100
1000010
A =
Amy Bob
Clara
Don Efaz
Finn
Bob Gal
Gal
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2A
0100000
1000010
0000001
0100000
0020000
1001000
0010100Finn
Amy Bob
Clara
Don Efaz
Clara Efaz
Gal
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Finn
Amy Bob
Clara
Don Efaz
3A
0000001
0010100
1000010
0000001
0200000
0020000
1101000
Don Finn Gal Gal
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Finn
Amy Bob
Clara
Don Efaz
0110001
1010111
1100011
0110001
1221000
1021100
2111110
32 AAA =
Gal
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This puzzle is reminiscent of the notion
of “six degrees of separation”, which
suggests that any two people in the
world are connected by a path of
acquaintances, whose average length
is six.
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Watts-Strogatz model
Average path length between two nodes in a random network
N=total nodes
K= acquaintances per node
KN ln/ln
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000,000,300N 30K
90% of the US population
Average path length= 4.3
5.19 =5.7
Acquaintances per person
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000,000,000,6N
90% of the World population
30K
Acquaintances per person
Average path length=
4.35.22
= 6.6
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Bibliography
Glaister, Paul. An application of Matrix Theory, The Mathematics Teacher, 85
(1992), pp. 220-223.
Lay, C. David. Linear Algebra. 4th ed.Massachusetts: Pearson Education, Inc.
2011.
Poole, David. Linear Algebra. 4th ed. Connecticut: Cengage Learning, 2014.
Duncan J. Watts & Steven H. Strogatz, Collective dynamics of ‘small-world’
networks, Nature, 393 (1998), 440-442.
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Thank You