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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

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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.

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

A. -2

B. -2b

C. 0

D. 2b

E. 2

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

Which is the following is equal to

does not exist

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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

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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

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7. If

what is the determinant of S^-1?

A. -2

B. -1/2

C. 1/4

D. 1/2

E. 2

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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.

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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

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10. What is the kernel of the linear transformation

A	B.	C.
D.	E.

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11. What is the inverse of the matrix

A	B.	C.
D.	E.

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