MA 305 Homework 6 - Bonus Homework

Due 11:59 pm on Sunday, May 14, 2000

This homework is a bonus homework and is not required. It is worth 30 points, or half of homework 5. Late submitions will not be accepted. Solutions will be posted before the final exam, but the homework will probably not be returned before it.

Calculations necessary for these problems may be done either by hand or with Maple. Label all problems and answers. Partial credit will only be given for included work. Solutions are to be submitted as ASCII text (.txt), Maple Worksheet (.mws), html (.html), postscipt (.ps), or portable document format (.pdf) file via WolfWare. (Please read the instructions for how to submit files.)

Questions about the assignment may be sent to the TA, Will Turner, at wjturner@math.ncsu.edu. Please note he will be out of town the four days before the homework is due. He will hold a few office hours during exam week, and they will be posted on the course web page.

  1. For the following, determine if the given function T is a linear transformation; if it is linear, find the matrix A such that T = fA.

    1. T(x,y,z) = (z-x,z-y)
    2. T(x,y) = (x-y,x+y,2x-3y)
    3. T:R2x2 -> R2x2 where
      T( [
      [      		 a
      b      		 c
      d      		 ]
      ]      		 ) = [
      [      		 2ab
      3cd      		 0
      0      		 ]
      ]

  2. For the matrix
    A = 
    [  5 1  2 0 ]
[ -1 0  2 0 ]
[  3 0 -2 0 ]
[  2 1  2 1 ]
    do the following.

    1. Find the characteristic polynomial of A in factored form.
    2. Find the eigenvalues of A.
    3. Find a basis for the eigenspaces of A.

  3. For this system of differential equations

    do the following:
    1. Find the general solution to the system of differential equations.
    2. Find the particular solution that satisfies the initial conditions u=7, v=4, and w=3 when t=0.