Extra Problems (back to full lecture notes)

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).

  1. Consider the following vector subspaces of R3:
    V
    =
    span {
     
    1
    0
    -2
     
    }
    W
    =
    span {
     
    4
    1
    2
     
    }
    You are in a spaceship positioned at:
    p
    =
     
    20
    15
    -30
     
    1. 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?
    2. 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?
  2. Consider the following underdetermined system:
     
    -2 4
    4 -8
     
     
    x
    y
     
    =
     
    -8
    16
     
    1. 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.
    2. Suppose that the cost of a solution vector v to the above system is defined to be ||pv|| where
      p
      =
       
      1
      5
       
      What is the solution to the above system that has the lowest cost?
  3. 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:
    f(v)
    =
     
    2 1
    4 2
     
    v
    g(v)
    =
     
    0.1 0.2
    0.4 0.3
     
    v
    v0
    =
     
    100
    200
     
    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?
  4. 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:
    f(v)
    =
     
    60 120
    20 40
     
    v
    g(v)
    =
     
    0.06 0.12
    0.02 0.04
     
    v
    v0
    =
     
    3
    1
     
    v100
    =
     
    300000
    100000
     

    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?