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Linear Algebra
Topic:

Vector Spaces

Multiple Choice Questions for Exam Preparation

1
...
The set of complex numbers ℂ = {𝑎 + 𝑏𝜄|𝑎, 𝑏 ∈ 𝐹} is a vector space over 𝐹 =
?
a) ℂ
b) ℝ
c) ℚ
d) All of above
3
...

a)
b)
c)
d)

d) Both (b) and (c)

Which statement is true?
If F is a subfield of E, then E is a vector space over F
The set 𝑆 = {(1,0), (2,0)} serves as basis of ℝ2
𝑀𝑚×𝑛 (ℝ) is a vector space over ℝ only if 𝑚 ≠ 𝑛
ℂ is not a vector space over ℚ

5
...


d) Y-axis

Let 𝑉 be a vector space over 𝐹, then any summand of the form
𝑎1 𝑣1 + 𝑎2 𝑣2 + ⋯ + 𝑎𝑛 𝑣𝑛 is called Linear Combination of 𝑣1 , 𝑣2 , … , 𝑣𝑛 ∈ 𝑉, 𝑎𝑖 , 𝑠 ∈ 𝐹 where?
a) 𝑣1 , 𝑣2 , … , 𝑣𝑛 are linearly dependent vectors
b) 𝑣1 , 𝑣2 , … , 𝑣𝑛 are unique vectors
c) 𝑣1 , 𝑣2 , … , 𝑣𝑛 are linearly independent vectors
d) Both (a) and (c)
7
...
If V = ℝ, 𝐹 = ℚ, then span(√5) is equal to?
a) ℚ
b) ℚ′
c) ℝ

d) None of these

9
...

The basis of a vector space 𝑉 over a field 𝐹 contains:
a) Every independent vector of 𝑉
b) Dependent and independent vectors
c) Finite number of vectors
d) Vectors which span the vector space 𝑽
11
...


If 𝑊 = {[

13
...

If 𝑊 = {(𝑎, 𝑏, 𝑐)|𝑎 = 𝑏 = 𝑐; 𝑎, 𝑏, 𝑐 ∈ ℝ} is a subspace of vector space 𝑉 = ℝ3 , then what will be
geometrical representation of W ?
a) Points in Space
b) Line passing through origin in space
c) Plane passing through origin in space
d) Surface passing through origin in space
15
...

Let V is a vector space over field F and for any subset S of V where 𝑆 = ∅ ,then
a) span(𝑆) = 𝑉
b) span(𝑺) = {𝟎}
c) span(𝑆) = {0} ∩ 𝑉
d) span(𝑆) = {0} ∪ 𝑉
17
...

Two vectors 𝑢 and 𝑣 in ℝ3 are linearly dependent iff:
a) They lie on same line through the origin
b) One is scalar multiple of other
c) Both vectors are equal

d) All of above
19
...

Consider a system 𝐴𝑋 = 0 of linear equations, where 𝐴 = 𝑀2×2 (𝐾), then which one is true?
a) 𝑿 ∈ 𝑲𝟐 , where K is any field
b) Zero vector does not belong to its solution set
c) The Solution set W of given system is subspace of ℝ𝑛
d) Given system is non-homogeneous system
21
...

a) 3

1
What will be the rank of matrix 𝐴 = [ 3
−1
3
b) 2

−2
1
−5
8

0 4
1 0 ]?
−1 8
2 −12
c) 1

d) 4

23
...

a) 1

If 𝑈 = {(𝑎, 2𝑎): 𝑎 ∈ ℝ} and 𝑉 = {(𝑏, 3𝑏): 𝑏 ∈ ℝ}, then dim(𝑈⨁𝑉) =?
b) 3
c) 2

d) 4

25
...

If 𝑉 = 𝑃2 (𝑡) and 𝑆 = {(𝑡 + 2), (𝑡 − 7), (𝑡 2 + 2𝑡 + 1)}, then what will be the coordinate vector [𝑣]𝑠
of 𝑣 = 𝑡 2 − 6𝑡 + 10 relative to 𝑆?
a) [3, −4,2]
c) [−2,7
...
2]
d) [−𝟓
...

If 𝑉 = ℝ3 and coordinate vector of 𝑣 = (5,2,7) is [𝑣]𝑠 = [5,2,7], then what is basis relative to 𝑣?
a) 𝑆1 = {(1, −1,0), (1,1,0), (0,1,1)}
c) 𝑺𝟐 = {(𝟏, 𝟎, 𝟎), (𝟎, 𝟏, 𝟎), (𝟎, 𝟎, 𝟏)}
b) 𝑆3 = {(−1,0,1), (0,0,1), (1,1,0)}
d) 𝑆4 = {(0,0,0), (1,1,0), (1,0,1)}
28
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What will be the dimension of subspace spanned by coordinate vectors:
[𝐴] = [1,2,6,0,0,0], [𝐵] = [2,3,4,1, −1, −3], [𝐶 ] = [2,0,7,3,4, −2]
a) 1
b) 2
c) 3
30
...
Then which mapping is determined by 𝐴?
a) 𝑭𝑨 : 𝑲𝒏 × 𝑲𝒎
b) 𝐹𝐴 : 𝐾 𝑛 × ℝ
c) 𝐹𝐴 : ℝ × 𝐾 𝑚
31
...

Which one of the following statements is true?
a) Every linear mapping takes zero vector into the zero vector
b) In linear mapping 𝐹: 𝑉 → 𝑈, both V and U are vector spaces over same field K
c) 𝐹 (𝑥 ) = 𝑥 𝑛 is linear mapping if 𝑛 = 1
d) A linear mapping of the form 𝑭: 𝑽 → ℝ𝒏 is called Linear Transformation
33
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Let 𝐹: 𝑉 → 𝑈 be a Linear Mapping then which one is true?
a) 𝐾𝑒𝑟𝐹 = {𝑣𝜖𝑈: 𝐹 (𝑣) = 0}
c) Kernal of 𝐹 is a subspace of 𝑈
b) Image of 𝐹 is a subspace of 𝑉
d) None of these
35
...

Which one of the following integers can be dimension of an algebra 𝐴(𝑉) of linear map?
a) 32
c) 49
b) 28
d) 88
37
...

Which one of the following linear operators is not invertible?
a) 𝐹: ℝ2 → ℝ2 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝐹 (𝑥, 𝑦) = (2𝑥 + 𝑦, 3𝑥 + 2𝑦)
b) 𝑮: ℝ𝟐 → ℝ𝟐 𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝒃𝒚 𝑮(𝒙, 𝒚) = (𝒙 + 𝒚, 𝒙 − 𝟐𝒚, 𝟑𝒙 + 𝒚)
c) 𝐻: ℝ2 → ℝ2 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝐻 (𝑥, 𝑦) = (𝑦, 𝑥)
d) Both (a) and (b)
Which one of the following 2 × 2 matrix maps (1,3)𝑇 and (1,4)𝑇 into (−2,5)𝑇 and (3, −1)𝑇 ?
13 −2
−𝟏𝟕 𝟓
a) 𝑨 = [
]
c) 𝐶 = [
]
21 7
𝟐𝟑 −𝟔
1 −1
23 −9
b) 𝐵 = [
]
d) 𝐷 = [
]
11 4
0 2

39
...


Consider two basis of ℝ3 :
𝑆 = {(1,2), (0,1)} & 𝑆 ′ = {(2,1), (−1,3)}, then what will be transition matrix from S to

S’?
−1 2
a) 𝑃 = [
]
0 −3
21 11
b) 𝑃 = [
]
8
4

1 2
]
4 −6
𝟐 −𝟏
d) 𝑷 = [
]
−𝟑 𝟓
c) 𝑃 = [

41
...

Consider 𝐿: ℝ2 → ℝ2 defined by as L is reflection of ℝ2 about 𝑦 = 𝑥
...

Let 𝐹 (𝑥, 𝑦) = (2𝑥 + 𝑦, 𝑥 − 3𝑦) and 𝑆 = {(1, −2), (2, −5)}, then for 𝑣 = (5, −7) which one is
[𝐹(𝑣)]𝑠 ?
a) (−55,33)𝑇
c) (𝟑, 𝟐𝟔)𝑻
b) (8, −6)𝑇
d) (−7, −5)𝑇
44
...

Consider a linear mapping : ℝ𝑛 → ℝ𝑚 , then which statement is true?
a) 𝑛𝑢𝑙𝑙𝑠𝑝(𝐴) = (𝑟𝑜𝑤𝑠𝑝(𝐴))⊥
b) For 𝑤 ∈ 𝑟𝑜𝑤𝑠𝑝(𝐴) and 𝑣 ∈ 𝑛𝑢𝑙𝑙𝑠𝑝(𝐴), we get 𝑤

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