MA 405 Homework 3 DUE: April 13, in class

This assignment will be turned in on paper.

If you use Maple to solve some of the questions, then simply print your worksheet out and attach to it the problems that you did by hand.

Please clean up your Maple worksheets and clearly label the problems. Remember you can omit output on commands by using a colon e.g. with(LinearAlgebra):

1.

Let

a) Find a basis for the row space of the matrix A.

b) Find a basis for the column space of the matrix A.

c) Find a basis for the nullspace of A.

d) (Existence): For every , is there at least one solution x to Ax=b? Please explain.

e) (Uniqueness): Is there at most one solution x for every to the equation Ax=b? Please explain.

2. Which of the following subsets of constitute a subspace of ? Prove or give a counter example.

a)

b)

3. Let and

a) Find , , and

b) Find, , , and . Where is the standard Euclidean norm.

c) Find the angle between u and v.

d) Find a basis for the orthogonal complement,, where .

4. Let

a) Find the projection of b onto from problem 3.

b) Verify that the error is orthogonal to u ,v.

c) Show the b can be expressed as b=p+q where p is in the column space and q is perpendicular to that space. In what space does q reside?