1.

Let

Graph ( A) in the plane.

2.

1. Fill in the missing steps in Example 3.16.

2. Let A by the matrix in Example 3.16. Compute the solution set of the equation A ^T w = 0, and show that the result is (3.8).

3.

For each of the following matrices A, determine if Ax = b has a unique solution for each b, that is, determine if A is nonsingular. For each matrix A which is singular, find a vector b such that Ax = b has a solution and a vector c such that Ax = c does not have a solution.

4.

For each of the following matrices A, determine if Ax = b has a unique solution for each b, that is, determine if A is nonsingular. For each matrix A which is singular, find a vector b such that Ax = b has a solution and a vector c such that Ax = c does not have a solution.

5.

Suppose A ? R ^{n n} and b ? R ⁿ, b ? 0. Is the solution set of the equation Ax = b,

a subspace of R ⁿ? Why or why not?

6.

Let D : C ¹[ a, b] ? C[ a, b