Mathematics I, Vector Analysis, Differential Equation, Fourier Series and Integral Transforms

Multiple Choice Questions 29 Pages

Mahatma Gandhi University

BSc Computer Science

MPC

Contributed by

Manpreet Prasad Chaudhuri

BSC CS Complimentary I – Mathematics I
,
Vector Analysis, Differential
Equation, Fourier Series and Integral Transforms
For Off Campus BSc cs Programme

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 _, _
∈
F, which of the
equation is correct:
(A) (_ + _) x = _x . _x
(B) _ (x + y) = _x + _y
(C) (_ + _) x = _x ⋃ _x
(D) (_ + _) x = _x _ _x

3. In any vector space V (F), which of the following results is correct?
(A) 0 . x = x
(B) _ . 0 = _
(C) (–_)x = –(_x) = _(– x)
(D) None of the above

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

7. If A and B are two subspaces of a vector space V(F), then
(A)
(B)
Page 1
(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) {(2_ + _, 0, _, _) | _, _
∈
R}
(B) {(2__+_, 0, _, _) | _, _
∈
R}
(C) {(2__ – _, 0, _, _) | _, _
∈
R}
(D) {(2_ – _, 0, _, _) | _, _
∈
R}

9. If V is said to form a vector space over F for all x, y
∈
V and _, _
∈
F, which of the
equation is correct:
(A) (__) x = _ (_x)
(B) (_ + _) x = _x . _x
(C) (_ + _) x = _x ⋃ _x
(D) (_ + _) x = _x _ _x

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
(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)

Page 2
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
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
Page 3
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; _, _
∈
F, then a
mapping T : V _ U is called a homomorphism or a linear transformation if
(A)
T(_x + _y) = _T(x) . _T(y)
(B)
T(_x + _y) = _T(x) + _T(y)
(C)
T(_x + _y) = _T(x) - _T(y)
(D)
T(_x + _y) = _T(y) + _T(x)

27. In any vector space V (F), which of the following results is correct?
(A)
0 . x = 0
(B)
_ . 0 = 0
(C)
(_ – _)x = _x – _x, _, _
∈
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 _, _
∈
F, which of the
equation is correct:
(A) (_ + _) x = _x + _x
(B) (_ + _) x = _x . _x
(C) (_ + _) x = _x ⋃ _x
(D) (_ + _) x = _x _ _x

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
Page 4
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)}
(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

Page 5
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)

40. If V = R4(R) and S = {(2, 0, 0, 1), (– 1, 0, 1, 0)}, then
(A) L(S) = {(2_ + _, 0, _, _) | _, _
∈
R}
(B) L(S) = {(2_ ⊕ _, 0, _, _) | _, _
∈
R}
(C) L(S) = {(2__, 0, _, _) | _, _
∈
R}
(D) L(S) = {(2_ – _, 0, _, _) | _, _
∈
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 _, _ real numbers, then which of the following is correct?
(A)
(B) = __
(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

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|
Page 6
(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)

1. Which of the following equation is correct in terms of linear transformation where T : V
_ W and x, y
∈
V, _, _
∈
F and V and where W are vector spaces over the field F.
(A) T(_x +_y) = _T(x) + _T(y)
(B) T(_x +_y) = _T(x) + _T(y)
(C) T(_x +_y) = _T(y) + _T(x)
Page 7
(D) T(_x +_y) = _T(x) . _T(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) _(T
1
T
2
) = (_T
1
)T
2
= T
1
(_T
2
) where _
∈
F
(B) _(T
1
T
2
) = _T
2
= _T
1
where _
∈
F
(C) _(T
1
T
2
) = _T
1
= (_T
2
) where _
∈
F
(D) _(T
1
T
2
) = _(T
1
+T
2
) = T
2
(_T
1
) where _
∈
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}
(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
Page 8
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

14. If T, T
1
, T
2
be linear transformations from V _ W, S, S
1
, S
2
from W _ U and K, K
1
, K
2
from U _ Z where V, W, U, Z are vector spaces over a field F then
(A) K(ST) = KST
(B) K(ST) = (KS)T
(C) K(ST) = KS
(D) K(ST) = ST

15. If T
1
, T
2
Page 9
∈

Hom (V, W) then
(A) r(_T
1
) = r(T
1
) for all _
∈
F, _ _ 0
(B) r(_T
1
) = r_ for all _
∈
F, _ _ 0
(C) r(_T
1
) = T
1
for all _
∈
F, _ _ 0
(D) None of the above

16. If T
1
, T
2
∈

Hom (V, W) and r(T) means rank of T then
(A) | r(T
1
) – r(T
2
) | = r(T
1
+ T
2
) = r(T
1
) + r(T
2
)
(B) | r(T
1
) – r(T
2
) | ≥ r(T
1
+ T
2
) ≥ r(T
1
) + r(T
2
)
(C) | r(T1) – r(T
2
) | _ r(T
1
+ T
2
) _ r(T
1
) + r(T
2
)
(D) | r(T
1
) – r(T
2
) | < r(T
1
+ T
2
) < r(T
1
) + r(T
2
)

17. Let T : V _ W and S : W _ U be two linear transformations. Then
(A) (ST)
–1
= T
–1
T
–1
(B) (ST)
–1
= T
–1
T
(C) (ST)
–1
= T
–1
S
–1
(D) None of the above

18. T be a linear operator on V and let Rank T
2
= Rank T then
(A) Range T ∩ Ker T = {0}
(B) Range T ∩ Ker T = {1}
(C) Range T ∩ Ker T = {2}
(D) Range T ∩ Ker T = {3}

19. A L.T. T : V _ W is called non-singular if
(A) Ker T = _
(B) Ker T = {0}
(C) Ker T = {1}
(D) Ker T = {2}

20. If T be a linear operator on R
3
, defined by T(x1, x2, x3) = (3x1, x1– x2, 2x1 + x2 + x3)
and (z1, z2, z3) be any element of R
3
then
(A) T
–1
(z1, z2, z3) = 0
(B) T
–1
(z1, z2, z3) = _
(C) T
–1
(z1, z2, z3) = 1
(D)
Ans: (D)

21. If T : V _ V is a L.T., such that T is not onto, and that there exists some 0 _ v in V such
that, T(v) = 0, then
(A) Ker T = {0}
(B) Ker T = _
(C) Ker T = {1}
(D) None of the above
Page 10

Mathematics I, Vector Analysis, Differential Equation, Fourier Series and Integral Transforms

Mathematics I, Vector Analysis, Differential Equation, Fourier Series and Integral Transforms

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