LINEAR ALGEBRA QUESTION BANK

LINEAR ALGEBRA QUESTION BANK

(1) (12 points total) Circle True or False: TRUE / FALSE: If A is any n ? n matrix, and In is the n ? n identity matrix, then InA = AIn = A.

TRUE / FALSE: If A, B are n ? n matrices, then the inverse of AB is A-1B-1.

TRUE / FALSE: If A, B are n ? 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?3 matrices, then their product AB is defined.

TRUE / FALSE: If A and B are both 2 ? 3 matrices, then the product ABT 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 ? 3 matrix. Then N ul(A) is a subspace of R2. (a) True (b) False

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

1

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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 3 1 -1 0

(9) What is the dimension of the null space Nul(A) of A = 0 0 4 2 0? 000 0 0

A.

1

B.

2

C.

3

D.

5

2 3 1 -1 0

(10) What is the dimension of the column space Col(A) of A = 0 0 4 2 0? 000 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?

000 0 0

A.

1

B.

2

C.

3

D.

5

LINEAR ALGEBRA QUESTION BANK

3

2 3 1 -1 0 (12) What is the dimension of the row space Col(AT ) of A = 0 0 4 2 0?

000 0 0

A.

1

B.

2

C.

3

D.

5

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LINEAR ALGEBRA QUESTION BANK

For questions 5 and 6: Suppose 1 -3 1 -2

A = 0 1 -1 1 1 -1 -1 0

and its reduced echelon form is 1 0 -2 1

U = 0 1 -1 1 00 0 0

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

1 -3 1 -2

A.

0 , 1 , -1 , 1

1 -1 -1 0

1 0

B.

0 , 1

0 0

1 -3

C.

0 , 1

1 -1

1 1 -2

D.

0 , -1 , 1

1 -1 0

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

1 0 1

A.

Span

-3

1

,

1 -1

,

-1 -1

-2

1

0

1 0

B.

Span

0

-2

,

1

-1

1

1

1 0 1

C.

-3

1

,

1 -1

,

-1 -1

-2

1

0

1 0

D.

0

-2

,

1

-1

1

1

LINEAR ALGEBRA QUESTION BANK

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(15) The matrix for a 90 counterclockwise rotation in the x-y plane is

A.

01 10

B.

0 -1 -1 0

C.

0 -1 10

D.

01 -1 0

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

L(p(t)) = 2p (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

A.

2 3 0

043

2 3 0

B.

0 2 6

002

3 2 0

C.

0 3 4

003

2 0 0

D.

3 2 0

062

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