APPLICATIONS I
Use pencil and show all work neatly organized on notebook paper. All computations may be done on the calculator.
1. Write


3 4
− 1 2


as a linear combination of the matrices


1 0
0 1


,


0 1
0 0


,


0 0
1 0


, and


0 0
1 1


.
2. Write the polynomial 3x
2 + 2x − 5 as a linear combination of the polynomials 1, 1 + x, and
1 + x + x .
3x + 2
2x − 1 x − 3 x + 2 a) Find the composite function L(T), simplifying algebraically.
b) Find the composite function T(L), simplifying algebraically.
c) Write the coefficient matrix of L d) Write the coefficient matrix of T e) Find LT and describe its relationship to L(T).
f) Find TL and describe its relationship to T(L).
L and T are in a special class of functions called linear fractional transformations. They must be of the form G = ax + b cx + d
, ad − bc ≠ 0.
The coefficients are complex numbers (remember that the real numbers are a subset of the complex numbers a+bi, where b = 0. It can be shown that the relationships you found above hold true in general for this class of functions.
For 3 points Extra Credit :
Given the linear fractional transformations L(x) = ax + b cx + d
and T(x) = ex + f gx + h
, find and simplify the composite functions L (T) and T (L). Let [L] =

 a b c d


and [T] =

 e f g h


.
Find LT and TL.
Show that LT is the coefficient matrix of L(T) and that TL is the coefficient matrix of T(L).
MATH 2000 - Applications I page 1
J. Ahrens 2/1/2004

4. Current flow in a simple electrical network can be described by a system of linear equations. A voltage source such as a battery forces a current to flow through the network. When the current passes through a resistor (such as a light bulb or motor), some of the voltage is “used up”. By
Ohm’s law, this “voltage drop” across a resistor is given by V = RI, where the voltage V is measured in volts, the resistance R in ohms (denoted by S ), and the current flow I in amperes (amps). The designated directions of the loop currents are arbitrary. If a calculated current turns out to be negative, then the actual direction of current flow is opposite to that chosen in the figure. If the current direction shown is away from the positive (longer) side of a battery around to the negative
(shorter) side, the voltage is positive; otherwise, the voltage is negative.
Kirchhoff’s Voltage Law : The sum of the RI voltage drops in one direction around a closed loop equals the sum of the voltage sources in the same direction around the loop.
5 Ω
Example: Determine each of the loop currents in this network. The direction of current flow through a source from to + is assumed positive. The direction of current flow in diagrams is chosen arbitrarily. When we write the equations, we consider voltage drop to be positive if we “go with the flow” and negative otherwise. For simplicity, we will consider
I n
to “originate” at the most recent junction.
75V
100V a
+
−
+
−
4 Ω
7 Ω
I
1
I
2
I
3 b
9 Ω
8 Ω
An analysis of the voltages and currents in the network yields the following equations:
Top loop: 75 = 5 + 9 + 4 = 14 + 4 I
2
I
2
I
3
I
3
I
3
Hmmm.... We have three variables, but only two equations...
Kirchhoff’s Current Law (paraphrased): Current toward a junction equals current away from it.
I
1
I
3
We will use an augmented matrix to solve the system: 

I
1
14 I
1
−
+ 4I
I
2
−
− +
I
3
= 0
4I
2
2
15I
3
=
=
75
100
I
1 this current is actually flowing in the opposite direction to that chosen in the diagram.




1
0
−
14 4 0 75
−
1 − 1 0
4 15 100




⇔




1 0 0 5.60
0 1 0 − .84
0 0 1 6.44




3 Ω
Determine the loop currents in this network.
Round answers to two decimal places.
+
75V
−
60V a
+
−
6 Ω
5 Ω
MATH 2000 - Applications I page 2
J. Ahrens 2/1/2004
I
1
I
2
4 Ω b
I
3
2 Ω

5. A path matrix counts the number of direct paths from one point to another. The rows represent the starting points, with the top row A being point A, etc. The columns represent the ending points, with the first column being point A, etc. For example, element a
12
= 1 because there is one path that goes directly from point A to point B, and a
31
= 0 because it is not possible to go directly from C to A.
We will use the following notation for this problem:
P1 matrix represents number of direct paths
P2 matrix represents number of two-length paths
P3 matrix represents number of two-length paths etc.
P 2 = (P1)(P1), P 3 = (P1)(P1) (P1), etc.
C
Example: P1 (direct paths): end A B C start A
B
C 



 0 1 1
1 0 1
0 1 0




 a) Write P2 (2-length paths): end A B C start A
B
C




1 1 1
1 1 1
 1 1 1




How do P2 and P 2 compare?
Compute P 2




1 1 1
1 1 1
 1 1 1




B b) Write P3 (3-length paths): end A B C start A
B
C




1 1 1
1 1 1
 1 1 1




How do P3 and P
2 compare?
Compute P 3




1 1 1
1 1 1
 1 1 1



 c) Could there be a pattern here????? Element a
23 should equal 3. What does it represent?
d) Compute P
4 and write in form end A B C start A
B
C




1 1 1
1 1 1
 1 1 1




Write all the possible 4-length paths from B to C, e.g. BABAC.
MATH 2000 - Applications I page 3
J. Ahrens 2/1/2004

Salt Lake City (L)
San Jose (J)
Cincinnati (C)
Knoxville (K)
Atlanta (A) Dallas/Fort Worth (D)
My daughter and her three adorable and brilliant daughters live in San Jose. There are NO direct flights between Knoxville and San Jose! Just getting from one city to the other at a reasonable time of day at an affordable price without touring every airport between here and there is well nigh impossible. After being stranded in O’Hare in below zero temperatures with no food or motel allowances, I’ve sworn off Chicago completely! My possible choices of airports are shown above.
a) Write P1 for Knoxville to San Jose: end A C D J K L start A
C
D
J
K
L




 



 1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1




 



 b) Calculate P 2 . What does it represent?
c) List all possible ways I can get from Knoxville to San Jose with only one stop.
d) List all possible ways I can get from San Jose to Knoxville with only one stop.
For 3 points Extra Credit : Use matrices to determine how many different ways there are to get from
Knoxville to San Jose without going through any airport twice.
MATH 2000 - Applications I page 4
J. Ahrens 2/1/2004