12. What are the eigenvalues of the matrix

 A. {-5, 1} B. {-1, 5} C. {-4,-2} D. {1, 3} E.{ 2+i sqrt(7) , 2 - i sqrt(7)

13. Let

For which of the vectors w does {u, v, w} form an orthogonal basis for R3 ?

 A. B C. D. E

14. Let be a basis for R4 and let L : R4 -> R2 be a linear transformation such that

What is the dimension of the kernel of L?

 A. 0 B. 1 C. 2 D. 3 E. 4

15. Let

What is the determinant of A?

 A. -4 B. -3 C. -2 D. -1 E. 0

16. Which of the following is a basis for the subspace of R3 generated by the vectors

 A B C D E Four vectors cannot span a subspace of R3

17. Let T: R2 -> R2 be the linear transformation defined by

What is the matrix associated with this transformation with respect to standard bases?

 A B C D E

18. Let

What is the entry on the first column of the third row of A-1 ?

 A. -3 B. -1 C. -1/2 D. 1 E. 2

19. Let

What is the determinant of A?

 A. -2 B. -1 C. 0 D. 1 E. 2

20. Let

Which of the following is a basis for the subspace of R3 generated by the eigenvectors of A corresponding to the eigenvalue 6:

 A B C D E

21. Let T be the linear transformation from R3 to R3 whose matrix with respect to the standard bases is

What is the value of (p, q, r)Transpose for which

 A B C D E

22. What is the value of the determinant

where

 A. -2 B. -1 C. 0 D. 1 E. 2

23. Let M be the subspace in R4 generated by the vectors

Let N be the subspace in R4 generated by the vectors

Let V be the inersection of the subspaces M and N. Which of the following pairs of vectors constitutes a basis for V?

 A B C D E

24. Let A be a square matrix whose characteristic polynomial is

What is the determinant of A?

 A. -1 B. -1/2 C. 0 D. 1/2 E. 1