Math 645, Linear Algebra

Very basic topic, but it takes time. For a large number of calculation problems, you don’t have to give too much process. But any question that needs a text answer must be detailed.

Math 645
Problem Set 2
Due Feb 19, 2020
Drill Exercises While the problems are not required, it is in your best interest to use these
problems as practice problems. The majority of mistakes that occur on exams are algebra and
speed of finishing a test. Practice will greatly increase your understanding and success in this
course. Do not use a calculator or computational tool when practicing, as one will not be permitted
on the exam.
Text §1.4 #s 1-12
Text §1.5 #s 1-6, 13-16
Text §1.7 #s 1-8, 11, 13, 14, 15-18
Text §1.8 #s 13-16
Required Problems:
Easier Problems – do by hand and/or just using reasoning:
A. Consider the following two vectors and determine if they are linearly independent. Why or why
not?




−1
2




 4 
 −8 
3
−6
B. Are the following vectors linearly independent? Why or why not? (Hint: you can do this by
inspection)






−1
2
1






 6 
 −5 
 1 
4
1
5
More Challenging Problems (feel free to use Matlab):
1. When the Russian ruble devalued dramatically in the 1990’s, a local businessman who had
dealings in Russia made a few key investments to protect his money by creating a closed economy.
In this case, he invested in a spaghetti noodle factory (output was pasta and a byproduct that
was essentially starch that could be fed to pigs or used as a binder in sausages), a pig farm (the
pigs happily ate the starch product and were then transformed into raw materials for…) a sausage
factory (that used the pig meat and the starch as a binder). The cool thing about this closed loop
was that it was somewhat self-sustaining without the need for cash, and any extra that could be
sold turned out to be very popular since pasta and sausages last a long time and could be purchased
as a hedge against inflation. At the simplest level, consider that the Noodle factory sold 10 % of its
output to the Sausage factory and 80 % of its output to the Pig farm, and kept the rest for its own
use. The Pig farm sold 30 % of its products to the Noodle factory and 50% of its product to the
sausage factory and retained the rest. The sausage factory sold 40 % of its output to the Noodle
factory and 40 % to the Pig farm.
(a) Construct an exchange table for this economy.
(b) Write a system of equations that can be solved to find the prices at which each factory/farm’s
income matches its expenses.
(c) Turn the equations in (b) into an augmented matrix and solve it to find a set of equilibrium
prices (assuming the price for the sausage factory output is set at 100 units).
(Page 2 as well)
2. Text §1.7 # 21, 22. This is a true-false question, however, to get full credit you must correctly
explain why the answer is True or False.
3. Text §1.7 # 9, 12, 19.
4. Text §1.8 #s 29, 30, 26.
5. Text §1.9 #s 4, 6, 11, 15, 19, 23.