MA 305 Homework 2

Due 11:59 pm on Friday April 11, 2003

Calculations necessary for these problems may be done either by hand or with Maple. Solutions are to be

submitted as a ASCII text (.txt) or Maple Worksheet (.mws) (version 7 or lower), file via WolfWare.

If you use Microsoft word, please save your file as ascii text with a .txt extension before submitting the

file.

Incorrectly submitted files will result in NO CREDIT, so be sure the file you submit is readable in a text

editor or Maple. If you are using Windows, check text files using notepad to make sure it looks readable.

Show your work for full credit. This means show all of your maple commands or if done by hand show most of

the intermediate steps so we know how you arrived at your answer.

You must write up your own homework. You may work together in the formulation of your answers, but you

must type your own file(i.e. no copying and pasting either).

Put your name and the names of all the people you worked with at the beginning of your submission file.

Questions about the assignment may be sent to the TA, Lauren D'Elia at rldelia@math.ncsu.edu.

1. Let

[2 4 2]

A =[1 5 2]

[4 -1 9]

a. Without computing A^-1, prove that A is invertible.

b. Compute 3x3 matrices T and U such that TA=U and U is in row echelon form.

c. Write T as a product of elementary matrices.

2. Let V be the set of all ordered pairs of real numbers with addition defined by
(x1,x2) + (y1,y2) = (x1 + y1, x2 + y2)

and scalar multiplication defined by

m o (x1,x2) = (mx1,x2)

The scalar multiplication for this system is defined in an unusual way and consequently we'll use the symbol o in order to avoid confusion with the ordinary scalar multiplication of row vectors. Prove or disprove that V a vector space with these operations.

3. Let

[1 -2 -1 1]

A = [2 4 -3 0]

[1 2 1 5]

a.Find a basis for the column space of A.

b.Find a basis for the row space of A.

c.What is the rank, r, of A.

d.True or False, for any b in R3 there is a solution to Ax=b? Explain.

e.True or False, for any b in R3 there is at most one solution to Ax=b? Explain.

4. Let S = {(x1,x2,x3)^T | x1 = x2}. Prove that S is a subspace of R3.

Bonus:

What is a basis for the nullspace of A in #3? Does the dim N(A) = n-r, where n is the number of columns A?