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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