Final Exam Review

Scheduling Information:

Time: Monday, December 11, from 1:00-4:00pm
Location: CUC McConomy
Review Session: TBA

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
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 are the coordinates of a vector with respect to a basis?
  2. What is a coordinate vector with respect to a basis?
  3. Regardless of the vector space V, the column vectors are always in ___ for some __.
  4. 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 ___________.
  5. What is meant by a "change of basis matrix?"
  6. How can we find a change of basis matrix when V=R^n?
  7. For an arbitrary vector space, how can we find a change of basis matrix?
  8. What is the matrix for a linear transformation with respect to certain bases?
  9. What is the matrix of a linear transformation with respect to bases B and C? What does it do?
  10. How can you find the matrix for a linear transformation with respect to bases B and C?
  11. What are similar matrices?
  12. What do similar matrices have to do with changing coordinates?
  13. What are some things that similar matrices have in common?
  14. What is a diagonalization of a matrix?
  15. Does every matrix have a diagonalization?
  16. What does diagonalization have to do with bases and coordinates?
  17. What properties of a matrix determine whether it has a diagonalization?
  18. What is an orthogonal diagonalization?
  19. What property do all orthogonally diagonalizable matrices have in common?
  20. What can be said about the eigenvalues of a real symmetric matrix?
  21. What can be said about the eigenvalues of a symmetric matrix?
  22. What is an orthogonal diagonalization?
  23. What kinds of matrices have an orthogonal diagonalization?
  24. How can you find an orthogonal diagonalization of an orthogonally diagonalizable matrix?
 

Exercises:

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 - well, except for #1 and #5b which you can ignore. 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.