Exam #3 Review

Scheduling Information:

Time: Monday, November, 7:30-8:20 AM.
Location: CUC McConomy.
Review Session: Saturday, November 18, from 3:00-4:30 in WEH 7500.
 

Reading:

Linear Independence and Spanning Sets: Sections 2.3, 6.2.
Bases: Section 3.4
Subspaces and Dimension: Sections 3.5 and 6.2.
The Four Fundamental Subspaces: Section 3.5 and 5.2.
Dot Products and Orthogonality: Sections 1.2 and 5.1.
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.
 

Review Questions:

  1. What is the span of a set of vectors?
  2. What is a subspace of a vector space?
  3. How can you know whether a set is a subspace?
  4. What is the row space of a matrix?
  5. What is the column space of a matrix?
  6. What is the null space of a matrix?
  7. Is the span of a set of vectors always a subspace?
  8. What does it mean for vectors to be linearly independent? Linearly dependent?
  9. How can you determine whether a set is linearly independent or dependent?
  10. What is a basis for a vector space?
  11. What is the dimension of a vector space?
  12. How can you find a basis for the row space of a matrix? Its dimension?
  13. How can you find a basis for the column space of a matrix? Its dimension?
  14. How can you find a basis for the null space of a matrix? Its dimension?
  15. What is the rank of a matrix? The nullity? How are they related?
  16. What is the algebraic multiplicity of an eigenvalue?
  17. What is the geometric multiplicity of an eigenvalue?
  18. What is an inner product?
  19. What are some examples of an inner product?
  20. What is the length, or norm, of a vector?
  21. What is the angle between two vectors? How is it related to inner products?
  22. What does it mean for vectors to be orthogonal?
  23. What does it mean for a set of vectors to be orthonormal?
  24. What does it mean for two subspaces to be orthogonal?
  25. What subspace is orthogonal to row(A)?
  26. What subspace is orthogonal to col(A)?
  27. What is the projection of a vector onto a line?
  28. What is a formula for the projection of a vector v onto span(u)?
  29. What is the projection of a vector onto a subspace?
  30. What is a formula for the projetion of v onto col(A)?
  31. What is an orthogonal decomposition of a vector?
  32. What does orthogonal decomposition have to do with orthogonal projections?
  33. How can you express the orthogonal projection of a vector in terms of an orthogonal basis? An orthonormal basis?
  34. What is an orthogonal matrix?
  35. What is the inverse of an orthogonal matrix?
  36. What quantities are "preserved" when multiplying by an orthogonal matrix?
  37. What is the Gram-Schmidt process?
  38. What is the difference between the "Gram-Schmidt process" and the "Modified Gram-Schmidt process?"
 

Exercises:

Section 1.2 #3, 9, 17, 19, 35, 63, 65.
Section 2.3 #1, 3, 7, 9, 19, 21, 23, 25.
Section 3.5 #3, 5, 7, 11, 15, 17, 19, 21, 23, 25, 33, 39, 45, 47.
Section 5.1 #1, 3, 11, 13, 28, 35.
Section 5.2 #3, 5, 7, 9, 11, 17,
Section 5.3 #1, 3, 5, 7, 11.
Section 6.1 #25, 35, 37, 43, 61.
Section 6.2 #7, 17, 23
 
 

Old Exam Problems:

Here is a selection of old exam problems I've given in previous semesters.

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.