Project I Linear Algebra 210
Spring I, 2006/LAGCC
I. (Stoichiometry, Text reference: pp. 59-60, exercises #5-10, p. 63) Consider the chemical reaction
XH
2
O + YFe -> ZFe(OH)
3
+ WH
2
We recognize the compounds in the chemical equation, reading from left to right as water, iron, ferric trihydroxide (?), and hydrogen (gas). We wish to find the smallest positive integers X, Y, Z, and W such that the chemical equation is balanced.
Question 1. Explain briefly what it means for a chemical equation to be balanced. What concept or principle can be used to obtain a system of linear equations for the variables?
How many equations are there, and how is this related to the number of different atoms in the chemical reaction?
Question 2. Can such a “stoichiometric” system of linear equations be inconsistent?
Explain. What two possibilities remain? Which type is this?
Question 3. Once you have obtained the equations, solve the system by the standard method of row reduction/Gauss elimination. Does your matrix solution give X, Y, Z, and
W directly? Explain what if anything you must do in addition.
II. (Traffic Flow, Text reference: pp. 60-62, exercises #11-14 p. 64) The figure below shows traffic circulation and inflow/outflow from a two-block neighborhood with six intersections (called nodes) and five one-way streets (called arcs) connecting them.
200vph 400vph
400vph F A 200vph
x
5 x
4
x
1
800vph E B 100vph
x
6
x
3
x
2
1000vph D C 100vph
x
7
800vph 600vph

The numbers and variables in the diagram represent the number of vehicles per hour that pass along each arc. For example, every hour, 600 vehicles flow into node A from outside the two-block network and must head for Nodes B or F. The unknown traffic flow x
1 represents the number of vehicles per hour that travel from Node A to Node B. If x
1
is negative, the vehicles actually flow from B to A.
Question 1. What concepts or principles (see text reference) enable you to write down a system of linear equations for the variables x
1
through x
7
? How many equations should there be? Explain.
Question 2. Does the resulting system have a unique solution, no solution, or an infinite number of solutions? Explain.
Now proceed to solve the system equations by using elementary row operations to transform your augmented matrix to reduced echelon form. Express your solution in parametric vector form q + x i u + x j v
+ …+ x k w
 where q, u, v, …, w are vectors and x i
, x j
, …x k
are (scalar) free variables. Check that the vector q that you obtain is a particular solution to the system of equations that you solved. Also check that the vectors u
,…, w are solutions to the homogeneous linear system.
Question 3. Write a brief description of the traffic pattern represented by the vector u and the traffic pattern represented by vector v . What closed loops are traversed in each?
III. (Electric Networks, Text reference: pp. 95-97, exercises #5-8) Current flow in an electric network in terms of V (volts), R (resistance measured in ohms), and I (current measured in amperes or amps) can be described by a system of linear equations. The
“linearity” comes from two physical laws. The first is Ohm’s Law, V=RI, providing scalar multiplication. The second, which assures additivity consists of two parts:
Kirchoff’s first law states that the total current coming into a node (or junction) equals the total current leaving a node.
Kirchoff’s second law
states that the net RI voltage drop around a loop must equal the sum of the component RI voltage drops within the loop.
Consider the electric network (also known as a circuit diagram) below:

Here is an example of the use of the first law: at node A, we have the equation
I
1
-I
2
-I
3
=0. Here is an example of the second law: in the closed loop BDCB, there is no external voltage, so the RI sum must be 0, hence the equation
R
4
I
4
-R
5
I
5
+R
6
I
6
=0. Note that we are going around the loop clockwise, so that the middle quantity is negative because of the direction of current flow.
Question 1. Obtain the other equations corresponding to nodes B, C, and D (first law) and loops BACB and DCAD (second law). Note that the external voltage is 0 in BACB and V in DCAD. Pay close attention to the direction of current flow to get the right signs. There should be 7 equations in all. Now suppose that R
1
=1, R
2
=R
5
=4, R
3
=R
4
=2, R
6
=6,
R
7
=3, and V=180. Solve the system for I
1
-I
7
by Gaussian elimination/row reduction.

IV. (Reflective essay) You have used the same elementary linear algebra (use of elementary row operations to obtain reduced echelon form) to solve three different applied problems. What was different in the settings of chemical equations, traffic flows and electrical networks was the assumptions and thought processes that went into generating the linear equations. I want you to write a short typewritten essay (but minimum of a page!) comparing and contrasting the principles…laws that you used to generate the linear equations. For example, how were Kirchoff’s laws similar to the traffic flow “laws”? How were they different? How would “conservation” apply in
Kirchoff’s laws, and how would it apply to stoichiometry? Were all systems consistent?
Were all solutions unique? Please include in your essay, a description of how you obtained the linear system in each of the three cases. What determined the number of equations in each case? You will submit this essay by the Blackboard digital drop box; hang on to computations for now, let answers to questions fuel your essay …more about that on Tuesday, April 18. Enjoy the rest of the holiday.