Mathematics-II

Multiple Choice Questions 47 Pages

Mahatma Gandhi University

BSc Computer Science

MPC

Contributed by

Manpreet Prasad Chaudhuri

MGU-BSc - BCS - 202 -[Computer Science]-[Complimentary - III]-Second Semester-
Mathematics-II

Unit-1-Linear Algebra: Vector Spaces-MCQs

1. Addition of vectors is given by the rule
(A)
(a
1
, b
1
) + (a
2
, b
2
) = (a
1
+ a
2
, b
1
+ b
2
)
(B)
(a
1
, b
1
) + (a
2
, b
2
) = (a
1
+ b
1
, a
2
+ b
2
)
(C)
(a
1
, b
1
) + (a
2
, b
2
) = (a
1
+ b
2
, b
1
+ a
2
)
(D)
(a
1
, b
1
) + (a
2
, b
2
) = (a
1
+ a
2
+ b
1
+ b
2
)

2. If V is said to form a vector space over F for all x, y ∈ V and α, β ∈ F, which of the
equation is correct:
(A) (α + β) x = αx . βx
(B) α (x + y) = αx + αy
(C) (α + β) x = αx ⋃ βx
(D) (α + β) x = αx ∩ βx

3. In any vector space V (F), which of the following results is correct?
(A)

0 . x = x
(B) α . 0 = α
(C) (–α)x = –(αx) = α(– x)
(D) None of the above

4. If α, β ∈F and x, y ∈ W, a non empty subset W of a vector space V(F) is a subspace of
V if –
(A) αx + βy ∈ W
(B) αx - βy ∈ W
(C) αx . βy ∈ W
(D) αx / βy ∈ W

5. If L, M, N are three subspaces of a vector space V, such that M ⊆ L then
(A) L ∩ (M + N) = (L ∩ M) . (L ∩ N)
(B) L ∩ (M + N) = (L + M) ∩ (L + N)
(C) L ∩ (M + N) = (L ∩ M) + (L ∩ N)
(D) L ∩ (M + N) = (L ∩ M ∩ N)

6. Under a homomorphism T : V → U, which of the following is true?
(A) T(0) = 1
(B) T(– x) = – T(x)
(C) T(0) = ∞
(D) None of the above

Page 1
7. If A and B are two subspaces of a vector space V(F), then

(A)
(B)
(C) A + B = A ∩ B
(D) Both (A) and (B)

Ans: (A)

8. If V = R
4
(R) and S = {(2, 0, 0, 1), (– 1, 0, 1, 0)}, then L(S)
(A) {(2α + β, 0, β, α) | α, β ∈ R}
(B) {(2αβ+β, 0, β, α) | α, β ∈ R}
(C) {(2αβ – β, 0, β, α) | α, β ∈ R}
(D) {(2α – β, 0, β, α) | α, β ∈ R}

9. If V is said to form a vector space over F for all x, y ∈ V and α, β ∈ F, which of the
equation is correct:
(A) (αβ) x = α (βx)
(B) (α + β) x = αx . βx
(C) (α + β) x = αx ⋃ βx
(D) (α + β) x = αx ∩ βx

10. If V is an inner product space, then
(A) (0, v) = 0 for all v ∈ V
(B) (0, v) = 1 for all v ∈ V
(C) (0, v) = ∞ for all v ∈ V
(D) None of the above

11. If V be an inner product space, then
(A) || x - y || ≤ || x || + || y || for all x, y ∈ V
(B) || x + y || ≤ || x || + || y || for all x, y ∈ V
(C) || x + y || ≥ || x || + || y || for all x, y ∈ V
(D) || x - y || ≥ || x || + || y || for all x, y ∈ V

12. If V be an inner product space, then
(A) || x + y ||
2
+ || x – y ||
2
= 2 (|| x ||
2
- || y ||
2
)
(B) || x + y ||
2
+ || x – y ||
2
= 2 (|| x || + || y ||)
2

Page 2
(C) || x + y ||
2
+ || x – y ||
2
= 2 (|| x ||
2
+ || y ||
2
)
(D) || x + y ||
2
+ || x – y ||
2
= 2 (|| x + y ||)
2

13. In Cauchy-Schwarz inequality, the absolute value of cosine of an angle is at most
(A) 1
(B) 2
(C) 3
(D) 4

14. If A and B are two subspaces of a FDVS V then, dim (A + B) is equal to
(A) dim A + dim B + dim (A ∩ B)
(B) dim A – dim B – dim (A ∩ B)
(C) dim A + dim B – (dim A ∩ dim B)
(D) dim A + dim B – dim (A ∩ B)

15. If A and B are two subspaces of a FDVS V and A ∩ B = (0) then
(A) dim (A + B) = dim A ⋃ dim B
(B) dim (A + B) = dim A + dim B
(C) dim (A + B) = dim A ∩ dim B
(D) dim (A + B) = dim (A + B)

16. If V be an inner product space and x, y ∈ V such that x ⊥ y, then
(A)
|| x + y ||
2
= || x ||
2
+ || y ||
2

(B)
|| x + y ||
2
= || x ||
2
. || y ||
2

(C)
|| x + y ||
2
= || x ||
2
⋃ || y ||
2

(D)
|| x + y ||
2
= || x ||
2
∩|| y ||
2

17. If V be a finite dimensional space and W
1
,..., W
m
be subspaces of V such that, V = W
1
+
... + W
m
and dim V = dim W
1
+ ... + dim W
m
, then
(A) V = 0
(B) V = dimW
1
⊕ ... ⊕ W
m

(C) V = ∞
(D) V = W
1
⊕ W
2
+ ... + ⊕ W
m

18. If V is a finite dimensional inner product space and W is a subspace of V, then
(A) V = W

. W
⊥

(B) V = W

+ W
⊥

(C) V = W

⊕ W
⊥

(D) V = W

∩ W
⊥

19. If W is a subspace of a finite dimensional inner product space V, then
(A)
(W
⊥
)
⊥
= W
(B)
(W
⊥
)
⊥
≠ W
(C)
(W
⊥
)
⊥
≤ W
(D)
(W
⊥
)
⊥
≥ W
Page 3
20. If W
1
and W
2
be two subspaces of a vector space V(F) then
(A) W
1
+ W
2
= {w
1
+ w
2
| w
1
∈ W
1
, w
2
∈ W
2
}
(B) W
1
+ W
2
= {w
1
. w
2
| w
1
∈ W
1
, w
2
∈ W
2
}
(C) W
1
+ W
2
= {w
1
∩ w
2
| w
1
∈ W
1
, w
2
∈ W
2
}
(D) W
1
+ W
2
= {w
1
⋃ w
2
| w
1
∈ W
1
, w
2
∈ W
2
}

21. If {w1,..., wm} is an orthonormal set in V, then for all v ∈ V is
(A)
Greater than or equal to || v ||
2

(B)
Less than or equal to || v ||
2

(C)
Greater than || v ||
2

(D)
Less than || v ||
2

22. If W is a subspace of V and v ∈ V satisfies (v, w) + (w, v) ≤ (w, w) for all w ∈ W
where V is an inner product, then
(A) (v, w) = ∞
(B) (v, w) = 1
(C) (v, w) = 2
(D) (v, w) = 0

23. If S
1
and S
2
are subsets of V, then:
(A) L(L(S
1
)) = L(S
1
)
(B) L(L(S
1
)) = L(S
2
)
(C) L(L(S
1
)) = L(V)
(D) L(L(S
1
)) = L(S
1
.S
2
)

24. If V be an inner product space and two vectors u, v ∈ V are said to be orthogonal if
(A) (u, v) = 1 ⇔ (v, u) = 1
(B) (u, v) ≠ 0 ⇔ (v, u) ≠ 0
(C) (u, v) = 0 ⇔ (v, u) = 0
(D) (u, v) = ∞ ⇔ (v, u) = ∞

25. A set {u
i
}
i
of vectors in an inner product space V is said to be orthogonal if
(A) (u
i
, u
j
) = 0 for i ≠ j
(B) (u
i
, u
j
) = 1 for i ≠ j
(C) (u
i
, u
j
) = ∞ for i ≠ j
(D) (u
i
, u
j
) = 2 for i ≠ j

26. If V and U be two vector spaces over the same field F where x, y ∈ V; α, β ∈ F, then a
mapping T : V → U is called a homomorphism or a linear transformation if
(A)
T(αx + βy) = αT(x) . βT(y)
(B)
T(αx + βy) = αT(x) + βT(y)
(C)
T(αx + βy) = αT(x) - βT(y)
Page 4
(D)
T(αx + βy) = αT(y) + βT(x)

27. In any vector space V (F), which of the following results is correct?
(A)
0 . x = 0
(B)
α . 0 = 0
(C)
(α – β)x = αx – βx, α, β∈ F, x ∈ V
(D)
All of the above

28. If V is said to form a vector space over F for all x, y ∈ V and α, β ∈ F, which of the
equation is correct:
(A) (α + β) x = αx + βx
(B) (α + β) x = αx . βx
(C) (α + β) x = αx ⋃ βx
(D) (α + β) x = αx ∩ βx

29. The sum of two continuous functions is ________________.
(A) Non continuous
(B) Continuous
(C) Both continuous and non continuous
(D) None of the above

30. A non empty subset W of a vector space V(F) is said to form a subspace of ___ if W
forms a vector space under the operations of V.
(A) V
(B) F
(C) W
(D) None of the above

31. If S
1
and S
2
are subsets of V, then:
(A) L(S
1
⋃ S
2
) = L(S
1
) + L(S
2
)
(B) L(S
1
⋃ S
2
) = L(S
1
) . L(S
2
)
(C) L(S
1
⋃ S
2
) = L(S
1
) ⊕ L(S
2
)
(D) L(S
1
⋃ S
2
) = L(S
1
) ∩ L(S
2
)

32. To be a subspace for a non empty subset W of a vector space V (F), the necessary and
sufficient condition is that W is closed under __________________.
(A) Subtraction and scalar multiplication
(B) Addition and scalar division
(C) Addition and scalar multiplication
(D) Subtraction and scalar division

33. If V = F
2
2
, where F
2
= {0, 1} mod 2 and if W
1
= {(0, 0), (1, 0)},W
2
= {(0, 0), (0, 1)},W
3

= {(0, 0), (1, 1)} then W
1
∪ W
2
∪ W
3
is equal to
(A) {(0, 0), (1, 0), (0, 1), (1, 1)}
Page 5
(B) {(1, 0), (1, 0), (1, 1), (1, 1)}
(C) {(0, 1), (1, 1), (0, 1), (1, 1)}
(D) {(0, 0), (1, 1), (1, 1), (1, 0)}

34. If the space V (F) = F
2
(F) where F is a field and if W
1
= {(a, 0) | a ∈ F}, W
2
= {(0, b) |
b ∈ F}then V is equal to
(A) W
1
+ W
2

(B) W
1
⊕ W
2

(C) W
1
. W
2

(D) None of the above

35. If V be the vector space of all functions from R → R and V
e
= {f ∈ V | f is even}, V
o
=
{f ∈ V | f is odd}. Then V
e
and V
o
are subspaces of V and V is equal to
(A)
V
e
. V
o

(B)
V
e
+ V
o

(C)
V
e
⋃ V
o

(D)
V
e
⊕ V
o

36. L(S) is the smallest subspace of V, containing _________.
(A) V
(B) S
(C) 0
(D) None of the above

37. If S
1
and S
2
are subsets of V, then
(A) S
1
⊆ S
2
⇒ L(S
1
) ⊆ L(S
2
)
(B) S
1
⊆ S
2
⇒ L(S
1
) ∩ L(S
2
)
(C) S
1
⊆ S
2
⇒ L(S
1
) ⋃ L(S
2
)
(D) S
1
⊆ S
2
⇒ L(S
1
) ⊕ L(S
2
)

38. If W is a subspace of V, then which of the following is correct?
(A) L(W) = W
(B) L(W) = W
3

(C) L(W) = W
2

(D) L(W) = W
4

39. If S = {(1, 4), (0, 3)} be a subset of R2(R), then
(A) (2, 1) ∈ L(S)
(B) (2, 0) ∈ L(S)
(C) (2, 3) ∈ L(S)
(D) (3, 4) ∈ L(S)

Page 6
40. If V = R4(R) and S = {(2, 0, 0, 1), (– 1, 0, 1, 0)}, then
(A) L(S) = {(2α + β, 0, β, α) | α, β ∈ R}
(B) L(S) = {(2α ⊕ β, 0, β, α) | α, β ∈ R}
(C) L(S) = {(2αβ, 0, β, α) | α, β ∈ R}
(D) L(S) = {(2α – β, 0, β, α) | α, β ∈ R}

41. In dot or scalar product of two vectors which of the following is correct?
(A)

(B) = 0
(C) = 1

(D) None of the above

Ans: (A)

42. If are vectors and α, β real numbers, then which of the following is correct?
(A)
(B) = αβ
(C) = 1
(D) = 0

Ans: (A)

43. If V is an inner product space, then
(A) (u, v) = 1 for all v ∈ V ⇒ u = 0
(B) (u, v) = 0 for all v ∈ V ⇒ u = 0
(C) (u, v) = ∞ for all v ∈ V ⇒ u = 0
(D) None of the above

Page 7
44. If V be an inner product space and v ∈ V, then norm of v (or length of v) is denoted by
(A) || v ||
(B)
(C) |v|
(D) None of the above

45. If V be an inner product space, then for all u, v ∈ V
(A) | (u, v) | = || u || || v ||
(B) | (u, v) | ≥ || u || || v ||
(C) | (u, v) | ≤ || u || || v ||
(D) | (u, v) | ≠ || u || || v ||

46. If two vectors are L.D. then one of them is a scalar ______ of the other.
(A)
Union
(B)
Subtraction
(C)
Addition
(D)
Multiple

47. If v
1
, v
2
, v
3
∈ V(F) such that v
1
+ v2 + v
3
= 0 then which of the following is correct?
(A) L({v
1
, v
2
}) = L({v
1
, v
3
})
(B) L({v
1
, v
2
}) = L({v
2
, v
2
})
(C) L({v
1
, v
2
}) = L({v
2
, v
3
})
(D) L({v
1
, v
2
}) = L({v
1
, v
1
})

48. The set S = {(1, 2, 1), (2, 1, 0), (1, – 1, 2)} forms a basis of
(A) R
3

(R)
(B) R
2

(R)
(C) R

(R)
(D) None of the above

49. If V is a FDVS and S and T are two finite subsets of V such that S spans V and T is L.I.
then
(A) 0 (T) = 0 (S)
(B) 0 (T) ≤ 0 (S)
(C) 0 (T) ≥ 0 (S)
(D) None of the above

50. If dim V = n and S = {v1, v2,...,vn} is L.I. subset of V then
(A) V ⊇ L(S)
(B) V ⊆ L(S)
(C) V ⊂ L(S)
(D) V ⊃ L(S)

Page 8
Unit-2-Linear Transformation-MCQs

1. Which of the following equation is correct in terms of linear transformation where T : V
→ W and x, y ∈ V, α, β ∈ F and V and where W are vector spaces over the field F.
(A) T(αx +βy) = αT(x) + βT(y)
(B) T(αx +βy) = βT(x) + αT(y)
(C) T(αx +βy) = αT(y) + βT(x)
(D) T(αx +βy) = αT(x) . βT(y)

2. If T : V → W be a L.T, then which of the following is correct
(A) Rank of T = w(T)
(B) Rank of T = v(T)
(C) Rank of T = r(T)
(D) None of the above

3. If T, T
1
, T
2
be linear operators on V, and I : V → V be the identity map I(v) = v for all v
(which is clearly a L.T.) then
(A) α(T
1
T
2
) = (αT
1
)T
2
= T
1
(αT
2
) where α ∈ F
(B) α(T
1
T
2
) = αT
2
= αT
1
where α ∈ F
(C) α(T
1
T
2
) = αT
1
= (αT
2
) where α ∈ F
(D) α(T
1
T
2
) = α(T
1
+T
2
) = T
2
(αT
1
) where α ∈ F

4. If T, T
1
, T
2
be linear operators on V, and I : V → V be the identity map I(v) = v for all v
(which is clearly a L.T.) then
(A)
T
1
(T
2
T
3
) = (T
1
T
3
)T
2

(B)
T
1
(T
2
T
3
) = (T
2
T
3
)T
1

(C)
T
1
(T
2
T
3
) = (T
1
T
2
)T
3

(D)
T
1
(T
2
T
3
) = (T
1
T
2
)

5. If T : V → W be a L.T, then which of the following is correct
(A) Nullity of T= w(T)
(B) Nullity of T = v(T)
(C) Nullity of T = r(T)
(D) None of the above

6. If T : V → W be a L.T, then which of the following is correct
(A) Rank T + Nullity T = dim V
(B) Rank T . Nullity T = dim V
(C) Rank T - Nullity T = dim V
(D) Rank T / Nullity T = dim V

7. If T : V → W be a L.T, then which of the following is correct
(A) Range T ∩ Ker T = {1}
Page 9
(B) Range T ∩ Ker T = {2}
(C) Range T ∩ Ker T = {3}
(D) Range T ∩ Ker T = {0}

8. If T : V → W be a L.T and if T(T(v)) = 0, then
(A) T(v) = 1, v ∈ V
(B) T(v) = ∞, v ∈ V
(C) T(v) = 2, v ∈ V
(D) T(v) = 0, v ∈ V

9.
If V and W be two vector spaces over the same field F and T : V
→
W and S : V
→
W be
two linear transformations then

(A)
(T + S)v = T(v) + S(v), v ∈ V
(B)
(T + S) v = T(v) . S(v), v ∈ V
(C)
(T + S)v = T(v) ⊕ S(v), v ∈ V
(D)
None of the above

10. If V, W, Z be three vector spaces over a field F and T : V → W, S : W → Z be L.T then
we can define ST : V → Z as
(A) (ST)v = ((ST)v)
(B) (ST)v = S(T(v))
(C) (ST)v = ((ST)v)
(D) (ST)v = (S(Tv))

11. If T, T
1
, T
2
be linear operators on V, and I : V → V be the identity map I(v) = v for all v
(which is clearly a L.T.) then
(A) IT = T
1

(B) IT = T
2

(C) IT = V
(D) IT = T

12. If T, T
1
, T
2
be linear operators on V, and I : V → V be the identity map I(v) = v for all v
(which is clearly a L.T.) then
(A) T(T
1
+ T
2
) = TT
1
+ TT
2

(B) T(T
1
+ T
2
) = T
1
+ T
2

(C) T(T
1
+ T
2
) = T(TT
1
+ TT
2
)
(D) T(T
1
+ T
2
) = TT
1
T
2

13.
If V and W be two vector spaces (over F) of dim m and n respectively, then

(A)
dim Hom (V, W) = mn
(B)
dim Hom (V, W) = m+n
(C)
dim Hom (V, W) = m⊕n
(D)
None of the above
Page 10

Mathematics-II

Mathematics-II

Download this file to view remaining 37 pages