MA 305 Homework 5

Due 11:59 pm on Sunday, April 30, 2000

This homework is worth 1.5 times the value of each of the previous four homeworks. 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.

  1. Let W = span{(1,1,1), (2,-1,1)}.

    1. Find a basis for the orthogonal complement of W.
    2. Describe the orthogonal complement of W geometrically.

  2. For the matrix
    A = 
    [ 1 3 1 2 ]
[ 1 4 1 1 ]
[ 2 1 2 1 ]
    do the following.

    1. Find a basis for the column space of A.
    2. Find a basis for the row space of A.
    3. Find the rank of A.

  3. Let u = (1, 2, 1) and v = (-1, 3, 2). Compute the following.

    1. The inner or scalar product of u and v: uT v.
    2. The Euclidean norm of u-v: ||u-v||.
    3. The angle between u and v.
    4. The projection of u onto v.

  4. Consider a surface with the general equation surface z = a x2 + b y + c and the following data.
    i xi yi zi
    1 0 0 2+1/10
    2 1 1 7-2/10
    3 -1 2 11-1/10
    4 1 -1 -1+2/10
    5 -1 1 7-1/10
    Using a least squares fit, do the following.

    1. Determine the constants a, b, and c and estimate z for (x,y) = (5, 10).
    2. Determine the matrix A and vectors x' and b' for this systems such that A x' = b'.
    3. Calculate the Euclidean norm of the residual, ||A x' - b||.

  5. 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> = 5 x1 y1 + x2 y2
    2. For x = (x1, x2) and y = (y1, y2),
      <x,y> = 5 x1 y1 - x2 y2
    3. For x = (x1, x2, x3) and y = (y1, y2, y3),
      <x,y> = x1 y1 + x2 y2
    4. For two polynomials f and g,
      <f,g> = f(0) g(0)

  6. Given vectors v1 = (1,1,2), v2 = (2,1,0), and v3 = (0,0,1), use the Gram-Schmidt process to find orthogonal vectors p1, p2, and p3 such that span{p1,p2,p3} = span{v1,v2,v3}.