LINEAR ALGEBRA QUESTION BANK

LINEAR ALGEBRA QUESTION BANK

(1) (12 points total) Circle True or False:

TRUE / FALSE: If A is any n ¡Á n matrix, and In is the n ¡Á n identity

matrix, then In A = AIn = A.

TRUE / FALSE: If A, B are n ¡Á n matrices, then the inverse of AB is

A?1 B ?1 .

TRUE / FALSE: If A, B are n ¡Á n matrices, then (A + B)?1 = A?1 +

B ?1 . (Hint: is this true for numbers?)

TRUE / FALSE: The Reduced Row Echelon Form of a matrix is unique.

TRUE / FALSE: If A and B are both 2¡Á3 matrices, then their product

AB is defined.

TRUE / FALSE: If A and B are both 2 ¡Á 3 matrices, then the product

AB T is defined.

(2) True or false: If a system of equations has more than one solution, it has

infinitely many solutions.

(a) True

(b) False

(3) True or false: If a system of equations is consistent, then it cannot have

any free variables.

(a) True

(b) False

(4) True or false: Let A be a 2 ¡Á 3 matrix. Then N ul(A) is a subspace of R2 .

(a) True

(b) False

(5) True or false: Let A be a 2 ¡Á 3 matrix. Then Col(A) is a subspace of R2 .

(a) True

(b) False

1

2

LINEAR ALGEBRA QUESTION BANK

(6) True or false: If V is a vector space of dimension d, and {v1 , . . . , vd } are d

different vectors in V , then they must form a basis.

(a) True

(b) False

(7) True or false: If V is a subspace of Rn , then every basis for V must have

the same number of vectors.

(a) True

(b) False

(8) True or false: If V is a vector space of dimension d, and {v1 , . . . , vd } are d

linearly independent vectors in V , then they must span V .

(a) True

(b) False

?

2

(9) What is the dimension of the null space Nul(A) of A = ?0

0

A.

1

B.

2

C.

3

D.

5

3

0

0

1

4

0

?1

2

0

?

0

0??

0

?

?

2 3 1 ?1 0

(10) What is the dimension of the column space Col(A) of A = ?0 0 4 2 0??

0 0 0 0 0

A.

1

B.

2

C.

3

D.

5

?

?

2 3 1 ?1 0

(11) What is the dimension of the left null space Nul(AT ) of A = ?0 0 4 2 0??

0 0 0 0 0

A.

1

B.

2

C.

3

D.

5

LINEAR ALGEBRA QUESTION BANK

?

2

(12) What is the dimension of the row space Col(AT ) of A = ?0

0

A.

1

B.

2

C.

3

D.

5

3

3

0

0

1

4

0

?1

2

0

?

0

0??

0

4

LINEAR ALGEBRA QUESTION BANK

For questions 5 and 6: Suppose

?

1 ?3

A = ?0 1

1 ?1

1

?1

?1

?

?2

1?

0

and its reduced echelon form is

?

1 0

U = ?0 1

0 0

?2

?1

0

?

1

1?

0

(13) Which of these is a basis for Col(A)?

A.

B.

C.

D.

?? ? ? ? ? ? ? ??

?3

1

?2 ?

? 1

?0? , ? 1 ? , ??1? , ? 1 ?

?

?

1

?1

?1

0

?? ? ? ??

0 ?

? 1

?0? , ?1?

?

?

0

0

?? ? ? ??

?3 ?

? 1

?0? , ? 1 ?

?

?

?1

1

?? ? ? ? ? ??

1

?2 ?

? 1

?0? , ??1? , ? 1 ?

?

?

1

?1

0

(14) Which of these is a basis for Col(AT )?

A.

B.

C.

D.

?? ? ? ? ? ??

1 ?

0

?

?

? 1

?

??3? ? 1 ? ??1??

?,? ?,? ?

Span ?

? ? ??1? ??1??

?

?

? 1

?

?

0

1

?2

?? ? ? ??

1

0 ?

?

?

?? ? ? ??

?

0? ?1?

?

Span ? ? , ? ?

?2

?1 ?

?

?

?

?

?

1

1

?? ? ? ? ? ??

1

0

1 ?

?

?

?? ? ? ? ? ??

?

?3

1

?1

? ?,? ?,? ?

? 1 ? ??1? ??1??

?

?

?

?

?

?2

1

0

?? ? ? ??

1

0 ?

?

?

?? ? ? ??

?

0

? ?,? 1 ?

?

?

?

?

?2

?1 ?

?

?

?

?

?

1

1

LINEAR ALGEBRA QUESTION BANK

(15) The matrix for

 a

0

A.

1



0

B.

?1

5

90? counterclockwise rotation in the x-y plane is

1

0



?1

0





0 ?1

1 0





0 1

?1 0

C.

D.

(16) Let L be the linear transformation from P2 to P2 given by

L(p(t)) = 2p0 (t) + 3p(t)

and let B = {1, t, t2 } be the standard basis for P2 . Then the coordinate

matrix A representing L with input and output basis B is:

?

?

3 0 0

?2 3 0?

A.

0 4 3

B.

?

2

?0

0

3

2

0

?

0

6?

2

C.

?

3

?0

0

2

3

0

?

0

4?

3

D.

?

2

?3

0

0

2

6

?

0

0?

2

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