In this assignment you will solve problems involving underdetermined systems and eigenvectors. All problems count
as extra credit; there is no penalty (in terms of your overall course grade) for not completing this assignment.
Please submit a single file a6.*
containing your solutions. The file extension may be anything you choose;
please ask before submitting a file in an exotic or obscure file format (plain text is preferred).

Consider the following vector subspaces of R^{3}:
You are in a spaceship positioned at:
 Suppose a transmitter's signal beam only travels in
two opposite directions along V. What is the shortest distance
your spaceship must travel to intercept the signal beam?
 Suppose the transmitter is also rotating around the
axis collinear with W. What is the shortest distance your
spaceship must travel to reach a position at which it can
hear the signal?

Consider the following underdetermined system:
 Define an isomorphism between the solution space W of the above equation and a parallel vector space V.
You must provide a definition of both the vector space and the isomorphism.
 Suppose that the cost of a solution vector v to the above system is defined to be p − v where
What is the solution to the above system that has the lowest cost?
 Suppose that vectors in R^{2} represent population quantities in two locations
(e.g., x is the number of people in the city and y is the
number of people in the suburbs), the linear transformation f:R^{2} → R^{2}
represents a change in the quantities over the course of one year if the economy is doing well, g:R^{2} → R^{2} represents
a change in the quantities over the course of one year if the economy is not doing well, and
v_{0} is the initial state:
The population state is initially v_{0}; after 302 years have passed, the total number of years with a good economy was 100
and the total number of years with a poor economy was 202. What is the state of the population distribution at this point?
 As above, let vectors in R^{2} be system state
descriptions that represent population quantities in two locations, and let f represent a change in
the quantities over the course of one year if the economy is doing
well, and let g represents a change in the quantities over the
course of one year if the economy is not doing well:
If the initial state is v_{0} and after 100 years
the state is v_{100}, during how many years
of this period was the economy doing well?