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 R3:
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 R2 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:R2 → R2
represents a change in the quantities over the course of one year if the economy is doing well, g:R2 → R2 represents
a change in the quantities over the course of one year if the economy is not doing well, and
v0 is the initial state:
The population state is initially v0; 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 R2 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 v0 and after 100 years
the state is v100, during how many years
of this period was the economy doing well?