Some Review Problems for Linear Algebra and Linear Transformations.

1. Suppose that

Which of these subsets of the vector space R4 are subspaces?

 A. None B. X only C. Y and Z only D. X, Y, and Z E. None of these

2. Let

Which of the follwing matrices represents the composition T1(T2) with respect to the standard bases for R2 .

 A. B. C. D. E.

3. If b is not 0 and ad-bc is not 0 then

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

4. Suppose T: R3 -> R3 is the linear transformation defined by

Which is the following is equal to

 A B C D E does not exist

5. Which of the vectors

are orthogonal to the vector

and have length less than 4?

 A. y only B. z only C. x and y only D. x and z only E. None of these

6. What is the value of c for which the vectors

are orthogonal?

 A. -5/7 B. -4/7 C. -3/7 D.-2/7 E. -1/7

7. If

what is the determinant of S-1?

 A. -2 B. -1/2 C. 1/4 D. 1/2 E. 2

8. Which of the following is a necessary and sufficient condition for the vector

to belong to the subspace of R3 generated by the vectors

 A. a - 3 b -4 c = 0 B. 2 a -4 b - 3 c = 0 C. 3a - 2 b - 2 c = 0 D. 4 a - 3 b -2 c = 0 E. 5 a - 7 b - c = 0.

9. Let T: R2 -> R2 be defined by

Which of the following is an equation of the line that is mapped onto itself by T?

 A. y-x = 0 B. 2 x + y = 0 C.-x+4y = 0 D. x + 2 y = 0 E. 4 x - y = 0

10. What is the kernel of the linear transformation

 A B. C. D. E.

11. What is the inverse of the matrix

 A B. C. D. E.