Practice Final for Math 33A

In addition to the problems below, it is helpful to look through the true/false problems at the end of each chapter in the book, as well as past midterm, practice midterm, quiz and homework problems. Please note: the absence of a particular subject from this list does not mean it will not be covered on the final.
  1. Find the matrix of the linear transformation T which is given by a rotation by 45o in the plane x+y+z=0 (counterclockwise as viewed from the point (1,1,1)) followed by the projection onto the z-axis. Find the rank of this matrix. Is it invertible?
  2. 2.3.29
  3. 2.4.51
  4. 3.1.39
  5. Find a basis for the kernel and the image of the matrix in 3.2.33
  6. 3.3.39
  7. 3.4.23
  8. 3.4.61
  9. 5.1.40--5.1.46
  10. 5.2.13
  11. 5.3.35
  12. 5.4.11
  13. 6.1.15
  14. 6.2.5
  15. 6.3.35
  16. 7.1.9
  17. 7.2.10
  18. 7.2.22
  19. 7.3.19
  20. 7.3.33
  21. 7.4.17
  22. 7.4.59
  23. Find a 3x3 matrix C so that C has exactly two eigenvalues, both of geometric multiplicity 1.
  24. 7.5.25. In this problem, also compute the images of the basis vectors e1,...,e4.
  25. 7.5.49
  26. If A is a 20x20 matrix with A100=0, find the characteristic polynomial fA.
Also do problems from sections 8.1--8.3 given on the last homework assignment.