The final exam will be on Wednesday, May 10, 2000, 10:30 in the morning (two hours). The eight problems will be as in Exam 1 and Exam 2, but the numbers may change. Neither notes nor calculators (except for #6) are allowed. In detail:

1. Solve a system with three unknowns and three equations using the LDU Decomposition of the matrix.

2. Given two explicit matrices A (2 by 2) and B (2 by 3), find the transpose of B, the inverse of A, the product of A and B, and so on.

3. Decide whether two given sets are subspaces.

4. Find the four fundamental subspaces of a given matrix, as well as bases for them and their dimensions.

5. Given are three vectors w1, w2, and v. Find the lengths of w1 and w2, their inner product, the distance between them, the angle between them, and the orthogonal complement of W=L(w1,w2). Is v in W perp? Is v in W? Find the orthogonal projection of v onto W, as well as the minimal distance from v to W.

6. This is a least square problem.

7. Compute the determinant of a given 4 by 4 matrix by a cofactor expansion along the fourth row.

8. Find all eigenvalues and corresponding eigenvectors for a given 2 by 2 matrix.