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
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 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
4
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
5
(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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