# Final Exam Review

## Scheduling Information:

Time: Tuesday, December 11, from 5:30-8:30pm
Location: POS A35
Review Session: Sunday, December 9, from 5:30-7:0pm, in WEH 8220

The Final Exam will be a cumulative exam. In preparing for the Final, you should also review the study guides for Exam #1, Exam #2 and Exam #3.

## Exam Format:

The exam will be given in two 80 minute parts. The first part will begin at 1:00 and must be completed by 2:20. Following that, there will be a 20 minute break before the second part begins at 2:40. The second part will conclude at 4:00.

Once each of the two parts begins, you may not leave the room until you have finished working on that part. Once you leave the room you will not be permitted to return to work on that part of the exam.

## Topics Covered:

Topics from Exam #1
Topics from Exam #2
Topics from Exam #3
Projections: Section 5.2 and 7.3.
The Orthogonal Decomposition Theorem: Section 5.2.
Orthogonal Matrices: Section 5.1.
The Gram-Schmidt Process: Section 5.3.
Least Squares Approximation: Section 7.3
Coordinates: Section 3.5 pp. 213-215.
Change of Basis: Section 6.3.
Matrix for a Linear Transformation: Section 6.6.
Similarity: Section 4.4.
Diagonalization: Section 4.4.
Orthogonal Diagonalization of Symmetric Matrices: Section 5.4.

## Review Questions:

1. What is the projection of a vector onto a line?
2. What is a formula for the projection of a vector v onto span(u)?
3. What is the projection of a vector onto a subspace?
4. What is a formula for the projetion of v onto col(A)?
5. What is an orthogonal decomposition of a vector?
6. What does orthogonal decomposition have to do with orthogonal projections?
7. How can you express the orthogonal projection of a vector in terms of an orthogonal basis? An orthonormal basis?
8. What is an orthogonal matrix?
9. What is the inverse of an orthogonal matrix?
10. What quantities are "preserved" when multiplying by an orthogonal matrix?
11. What is the Gram-Schmidt process?
12. What is the difference between the "Gram-Schmidt process" and the "Modified Gram-Schmidt process?"
13. What is a least squares solution of Ax=b?
14. What are the normal equations for Ax=b?
15. How can you find a least squares solution x* for Ax=b?
16. What do least squares solutions have to do with projections?
17. What are the coordinates of a vector with respect to a basis?
18. What is a coordinate vector with respect to a basis?
19. Regardless of the vector space V, the column vectors are always in ___ for some __.
20. The mapping that takes a vector v in V to it's coordinate vector with respect to a particular basis is a ______ ______________. In fact it is an ___________.
21. What is meant by a "change of basis matrix?"
22. How can we find a change of basis matrix when V=R^n?
23. For an arbitrary vector space, how can we find a change of basis matrix?
24. What is the matrix for a linear transformation with respect to certain bases?
25. What is the matrix of a linear transformation with respect to bases B and C? What does it do?
26. How can you find the matrix for a linear transformation with respect to bases B and C?
27. What are similar matrices?
28. What do similar matrices have to do with changing coordinates?
29. What are some things that similar matrices have in common?
30. What is a diagonalization of a matrix?
31. Does every matrix have a diagonalization?
32. What does diagonalization have to do with bases and coordinates?
33. What properties of a matrix determine whether it has a diagonalization?
34. What is an orthogonal diagonalization?
35. What property do all orthogonally diagonalizable matrices have in common?
36. What can be said about the eigenvalues of a real symmetric matrix?
37. What can be said about the eigenvalues of a symmetric matrix?
38. What is an orthogonal diagonalization?
39. What kinds of matrices have an orthogonal diagonalization?
40. How can you find an orthogonal diagonalization of an orthogonally diagonalizable matrix?

## Exercises:

Section 5.3 #1, 3, 5, 7, 11.
Section 7.3 #7, 15, 21, 23, 29, 37, 41.
Section 6.3 #3, 7, 15, 17.
Section 3.5 #3, 5, 7, 11, 15, 17, 19, 21, 23, 25, 33, 39, 45, 47, 51.
Section 6.6 #1, 3, 7, 13, 15, 17, 39.
Section 4.4 #1, 3, 5, 11, 13, 19, 23, 29, 33, 35.
Section 5.3 #13, 15, 19.
Section 5.4 #1, 5, 9, 15, 25, 27.

## Old Exam Problems:

Here are some old exam problems. They should be a pretty good guide to what you might expect to see on Monday's exam. This is not a complete final exam, I've tried to present problems that cover material from the last few weeks of the course.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. You'll have a chance to ask questions at the Review Session, and during your TA's office hours, too.

## Reference Tables:

I'll give you this reference table to use with your exam.