MA 305 Homework 4

Due 11:59 pm on Friday, April 7, 2000

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. Define the set P3 = {a+bx+cx2+dx3 : a,b,c,d real numbers}, the set of polynomials of degree 3 or less with real coefficients. Show that P3 is a vector space over the real numbers with the usual scalar multiplication and addition over the polynomials.

  2. For the vector space P3 defined above, do the following.

    1. Find a basis for P3.
    2. Give the dimension of P3.

  3. Find a basis for the right null space of the matrix
    A = 
    [ 1 3 1 2 ]
[ 1 4 1 1 ]
[ 2 1 2 1 ]

  4. Consider the vector space R4 of four-dimensional vectors, and the two vectors, 
    [ 1 ]
[ 1 ]
[ 2 ]
[ 1 ]
    		 and 
    [ 0 ]
[ 1 ]
[ 0 ]
[ 1 ]

    1. Show these two vectors are linearly independent.
    2. Extend these two vectors to a basis for R4.