MA-305 Homework 6

Due at 2:35 pm, Thursday, April 30, 1998


Calculations necessary for these problems may be done either by hand or with Maple. Solutions may be submitted in person in class, or you may email an ASCII text, Maple Text, or Maple Worksheet (.mws) to the TA, John Haws (jchaws@eos.ncsu.edu).

Remember: If you have a question, you may find the answer in the Forum.

  1. Consider the general equation for the following surface:
    z = ax2 + by + c
    and the following data:
    i
    1.
    2.
    3.
    4.
    5.     		xi
    0
    1
    -1
    1
    -1     		yi
    0
    2
    2
    -1
    -2     		zi
    3 - 1/10
    15 + 2/10
    15 - 3/5
    0 + 7/8
    -5 - 1/5

    Using a least squares fit, (i) determine the constants a, b, and c, and estimate z for (x,y) = (25,10); (ii) determine the matrix A and the vectors x' and b' for this system such that Ax' = b'; (iii) Calculate the Eucidean norm of the residual, ||Ax'-b||; (iv) determine a basis for the orthogonal complement of A; and (v) write the residual Ax'-b as a linear combination of vectors in the orthogonal complement of A.
    (Note: This problem will be worth 20 points.)

  2. Determine which of the following are inner products. Justify your answer. For those that are inner products, describe the corresponding norm.
    1. For x=(x1, x2) and y=(y1, y2),
      <x,y> = 10x1y1 + 4x2y2.
    2. For x=(x1, x2, x3) and y=(y1, y2, y3),
      <x,y> = x1y1 + x3y3.
    3. Let f and g be two polynomials of degree 2 or less, and <f,g> = f(-1)g(-1) + f(0)g(0) + f(1)g(1).

  3. For the following, determine if the given function T is a linear transformation. Justify your answer. If T is a linear transformation, the find the matrix A such that T = fA.
    1. T(x,y) = (y,x)
    2. T(x,y,z) = (z-x,z-y)
    3. T:M2x2 -> M2x2 where
      T( [
      [       		w
      y       		x
      z       		]
      ]       		) = [
      [       		w+z
      0       		0
      x+y       		]
      ]

(Bonus #1) For the following matrix:
A = [
[
[ 		-3
20
2 		1
3
-2 		-3
10
4 		]
]
]

  1. Find its characteristic polynomial (expressed as a product of linear factors).
  2. Find its eigenvalues.
  3. Find a basis for its eigenspace.
(Bonus #2) Find (i) the General Solution to the following system of differential equations, and (ii) the particular solution that satisfies the given intial conditions:
     du/dt=-u-3v
     dv/dt=-3u+7v
     u=4, v=7, when t=0.

Problems 2 and 3, and Bonus #2 from Elmentary Linear Algebra with Applications by Hill.

Bonus Problem #1 from Matrix Analysis and Applied Linear Algebra by Meyer.