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# This is Applied Math class from Stony Brook.

I actually dont know how to solve these problems with using computer coding.
Can you solve all of them or see the ways to solve them.

AMS 210 Homework #3 (75 points) Due Date: Thursday, Feb 28, 2013 For this homework, there will be less focus on by-hand exercises and more emphasis on seeing what one can do with linear systems of equations. Un- less speci ed otherwise, you are welcome to use computer algebra software, e.g. Sage, MATLAB, Mathematica, Maple, etc. He eron pp. 59-60 has a brief introduction for solving systems of equations using Sage or Maple (useful for Leontief/Input-Output type models). For dynamic problems, like the predator- prey problem or the economic model I presented, you are welcome to reference the Sage notebook that I presented in class. When you do solve using a com- puter, please please please explicitly state which computer algebra software. 1. (20 points) Gauss-Jordan Elimination The family of solutions of x - 2 z + w = 0 - x + y + 4 z = 0 - 2 x + 3 y + 10 z + w = 0 is a two dimensional plane in R 4 . Do this exercise by hand and show all of your work. (a) (5 points) Expressed as a parameterized set of vectors of the form { z ~ h 1 + w ~ h 2 : z,w R } . (b) (5 points) Similar as above but using x and y as parameters, i.e. the form { x ~ h 3 + y ~ h 4 : x,y R } . (Hint: Smart pivot selection will make this easier) (c) (10 points) Show that the two sets of vectors are equivalent, i.e. each member of the set in part (a) can be expressed as a member of the solution in part (b). Note that it is su cient to show that two vectors from one set are members of the other, but how do you extend this fact to show that every member of one set is a member of the other? 2. (25 points) Leontief Economic Model Suppose you are given a table which describes the industrial demand of a simple economy based on four goods/services: fuel, construction, transportation, steel. 1
Demands so many units of Fuel Const. Trans. Steel One unit of Fuel .3 .3 .2 .1 Const. .3 .3 .15 .2 Trans. .1 .1 0 .1 Steel 0 .1 .1 0 Table 1: Industrial Demand (a) (5 points) Set up a system of equations describing a Leontief Model for the economy, with supply of each good equal to the sum of its total industrial and consumer demand. Assume that customer demand is xed for 100 units of each good or service. (b) (5 points) Solve the system of equations using a computer algebra software of your choice. Include the code that you used to solve the system. (c) (15 points) We can also solve the above system iteratively , that is in a dynamic sense but where the time we used is not meant in a literal sense. Use the consumer demand for each good or service as an initial condition and see what the total demand (industrial plus consumer) is to produce this good. Include a table detailing the state of the supply at the tenth, twentieth, ftieth, and hundredth iteration. How close is the number of goods at the last iteration to the exact answer you got in part (b)? You may want to consult Example 2 in Section 1.2 of Tucker's book to see this iterative process in action. Again, please attach the code that you used to implement this task. 3. (25 points) Predator-Prey Model Let's consider the population model for rabbits and foxes. Normally we assume that rabbits reproduce at a rate proportional to their population and foxes die in proportion to their popu- lation. But we can change this model slightly to get other behavior. Sup- pose instead that rabbits do not reproduce but are instead re-introduced with a constant number of new rabbits at each time step, while simulta- neously the number of foxes is reduced by removing a constant number of foxes at each time step. (a) (5 points) Write a system of equations which relates the population at the next time step in terms of the current population. (Hint: b = d = 0 ) 2
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