Free Printable Vector Space-Subspace-Linearly Independent-Bases-Dimension

1.

The subset { (1,-2), (2,9), (-4,3 } of $R^2$ is

a)

Linearly dependent

b)

Linearly independent

c)

Basis

d)

None of the above

2.

The set { (1,-2,6), (5,-10,30) } is

a)

Linearly dependent

b)

Linearly independent

c)

Basis

d)

None of the above

3.

Dimension of set of 3 $\times$ 3 matrices with real entries is

a)

6

b)

9

c)

3

d)

0

4.

If W is subspace of vector space v then

a)

dim W = dim V

b)

dim w $<$ dim V

c)

dim w $>$ dim V

d)

dim w $\le$ dim V

5.

The dimension of subspace W = { (x,y,z) / x+y+z = 0 } of $R^3$ is

a)

1

b)

3

c)

2

d)

0

6.

Is the set { (4,5,3), (1,0,2), (0,0,0) } is basis of $R^3$

a)

Yes

b)

No

c)

Can not be determined

d)

None of the above

7.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

8.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

9.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

10.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

11.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

12.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

13.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

14.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

15.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

16.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

17.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

18.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

19.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

20.

a)

Option (a)

b)

Option (b)

c)

Option (c)

d)

Option (d)

21.

1. Let V be a vector space over a field F and $\alpha\in F$ and $u\in V$ . Which of the following statement is not correct?

a)

$\alpha u=\theta\Longrightarrow either\ \alpha=0\ or\ u=\theta$

b)

$\left|-1u\right|=\left|-1\right|u,\forall u\in V$

c)

$\alpha\theta=\theta$

d)

$0u=\theta$

22.

1. Let V be the vector space of all 2X2 matrices over R, Then, the set W, consisting of all matrices A for which $A^2=A$ ,

a)

is a subspace of V

b)

is not a subspace of V as it is not closed with respect to vector addition

c)

is not a subspace of V as it is not closed with respect to scalar multiplication

d)

Both (b) and (c)

23.

The zero vector in the vector space R⁴ is

a)

(0,0)

b)

(0,0,0)

c)

(0,0,0,0)

d)

none of these

24.

In vector space V(F), the set V contains

a)

Scalars

b)

Vectors

c)

Scalars and vectors

d)

None of these

25.

In vector space V(F), the set F contains

a)

Scalars

b)

Vectors

c)

Scalars and vectors

d)

None of these

26.

In a vector space V(F), binary operation is

a)

a) vector addition

b)

b) scalar multiplication

c)

c) both a and b

d)

d) None of these

27.

The linear span L(S) of any subset S of a vector space V(F) is a (a) of V(F).

28.

Is null set a vector space?

a)

Yes

b)

No

29.

Set of polynomials of degree 3 is a vector space. The statement is

a)

True

b)

False

30.

In a finite dimensional vector space V of dimension n, subset A of V has n number of linearly independent elements. Then

a)

A is generating set of V

b)

A is not a generating set of V

c)

Removing some elements from A can give generating set.

d)

Adding some elements to A can give basis

31.

Let V be vector space of polynomials of degree upto three. Let S be set of polynomials of degree 2. Is S a subspace of V?

a)

Yes

b)

No

32.

In a vector space of dimension 5, a set containing 6 elements is

a)

Linearly independent

b)

Linearly dependent

c)

Generating set

d)

Can not say anything about the set

33.

In a vector space of dimension 5, a subset A containing 6 elements is generating set. Then

a)

A is Linearly independent also.

b)

There is one element in A which can be written as a linear combination of others.

c)

A contains zero element

d)

Can not say anything about the set A

34.

Let $V$ be a vector space over the field $F$ . Let $W$ be a subset of $V$ . Then $W$ is a subspace of $V$ over the field $F$ if

a)

$W$ is a vector space over the field $F$

b)

$\forall\ x\ ,\ \ y\ \in W,\ x+y\in W\ and\ xy\in W$

c)

$\forall\ x\ ,\ \ y\ \in W\ and\ c\in F,\ x+y\in W\ and\ cy\in W$

d)

$\forall\ x\ ,\ \ y\ \in W\ and\ c\in F\ ,\ \ cx+y\in W\$

35.

The dimension of the vector space ℝⁿ over the field ℝ is.......

a)

ℝ

b)

0

c)

∞

d)

n

36.

A vector space is said to be finite dimensional vector space if the dimention of the vector space is infinite.

a)

True

b)

False

37.

If W₁ and W₂ are two subspace of vector space V(F). Then which of the following is false

a)

W₁ ∪ W₂ is a subspace of V(F)

b)

W₁ ∩ W₂ is a subspace of V(F)

c)

W₁ + W₂ is a subspace of V(F)

d)

W₁ ∪ W₂ is a not a subspace of V(F)

38.

If W is a subspace of a vector space V(F) and α, β ∈ W then for any scalar c ..........

a)

cα + β ∈ V

b)

cα + β ∈ W

c)

cα + β ∉ V

d)

cα + β ∉ W

39.

If W₁ and W₂ are two disjoint subspace of vector space V(F). Then.....

a)

W₁ ⋂ W₂ = ∅

b)

W₁ ⋃ W₂ = {0}

c)

W₁ ⋂ W₂ = {0}

d)

W₁ ⋃ W₂ = ∅

40.

$𝐷𝑖𝑚(𝑀_{2X3})=_{_{_{_{_{_.}}}}}$

a)

2

b)

3

c)

6

d)

5

41.

Is the vectors (1, 2, 3), (4, 5, 6) and (7, 8, 9) span the vector

space ℝ3.

a)

yes

b)

no

42.

Let 𝑉 = ℝ³. Is 𝑊 is a subspace of 𝑉 where W= {(𝑎, 𝑏, 𝑐): 𝑎 ≥ 0}.

a)

yes

b)

no

43.

Let V be a vector space then

a)

u+v=v+u

b)

1v=v

c)

$\alpha\left(\beta v\right)=\left(\alpha\beta\right)v$

d)

All of these

44.

Let V be a vector space, and let W be a subset of V. What does it mean when we say that is linearly independent?

a)

S is closed under both addition and scalar multiplication.

b)

The only way to write 0 as a linear combination of elements of S.

c)

All the elements of are distinct from each other.

d)

S has nullity zero.

45.

Let V be a vector space, and let S be a subset of V. What does it mean when we say that S spans V?

a)

Every vector in V can be expressed as a linear combination of vectors in S.

b)

The elements of S are all distinct from each other.

c)

Every vector in V has exactly one representation as a linear combination of vectors in S.

d)

S is a basis for V.

46.

If V is a vector spacr over F then $0.v=0\$ for $v\in V$

a)

True

b)

False

47.

Let V be a five-dimensional vector space, and let S be a subset of V which spans V . Then S

a)

Must have exactly five elements.

b)

Must consist of at least five elements.

c)

Must have infinitely many elements.

d)

Must have at most five elements.

48.

L(S) is a subspace of V.

a)

True

b)

False

49.

The dimension of V over F is ---------

a)

Number of elements in V

b)

Number of elements in F

c)

Number of elements in any basis of V over F.

50.

If V is finite dimensional then any two bases of V

a)

are in W

b)

are equal

c)

have the same number of elements

51.

Determine if the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9) are linearly independent.

a)

No, the vectors are collinear.

b)

Yes, the vectors are linearly independent.

c)

No, the vectors are orthogonal.

d)

No, the vectors are not linearly independent.

52.

What is the dimension of the vector space spanned by the vectors (1, 2, 3) and (4, 5, 6)?

a)

5

b)

2

c)

3

d)

0

53.

The set of all polynomial with degree less than or equal to n is a Vector Space

a)

True

b)

False

54.

The value of $k$ so that vectors $\left(1,-1,3\right),\left(1,2,-2\right),\left(k,0,1\right)$ are linearly independent.

a)

5/4

b)

1/4

c)

3/4

d)

7/4

55.

Let us consider $\left(1,1,2,4\right),\left(2,-1,-5,2\right),\left(1,-1,-4,0\right)\ and\ \left(2,1,1,6\right)$ vectors in $R^4$ , then

a)

they are linearly depenent

b)

they are linearly independent

c)

cannot be determined

d)

none of the above

56.

If $W_1\ and\ W_2$ are the finite dimensional subspaces of vector space $V$ , then

a)

$\dim\left(W_1+W_2\right)=\dim W_1+\dim\ W_2$

b)

$\dim\left(W_1+W_2\right)=\dim W_1+\dim\ W_2-\dim\left(W_1\cap W_2\right)$

c)

$\dim\left(W_1+W_2\right)+\dim\left(W_1\cap W_2\right)=\dim W_1+\dim\ W_2$

d)

$\dim\left(W_1+W_2\right)=\dim\left(W_1\cap W_2\right)$

57.

Let $V$ be the real vector space of all functions $f$ from $R\ into\ R\$ . Which of the following sets of functions are subspaces of $V$ ?

a)

all $f$ such that $f\left(x^2\right)=\ f\left(x\right)^2$

b)

all $f$ such that $f\left(0\right)=f\left(1\right)$

c)

all $f$ such that $f\left(-1\right)=0$

d)

all $f$ such that $f\left(3\right)=1+f\left(-5\right)$

58.

Let $V$ be a vector space over the field $F$ . Let $W_1\ ,\ W_2\ ,\ W_3$ are the subspaces of $V$ . Then

a)

$W_1\cup\left(W_2\cap W_3\right)$ is a subspace of $V$

b)

$W_1\cap\left(W_2\cap W_3\right)$ is a subspace of $V$

c)

$W_1\ \cup W_2$ is a subspace of $V$

d)

$W_2\ \cap W_3$ is a subspace of $V$

59.

Let $V$ be set of all pairs $\left(x,\ y\right)$ of real numbers and let $F$ be the field of real numbers, Define $\left(x,\ y\right)+\left(x_1,\ y_1\right)=\left(x+x_1,\ y+y_1\right)$ and $c\left(x,\ y\right)=\left(cx,\ y\right)$ , then

a)

undefinedis a vector space over the field of real numbers.

b)

undefinedis not a vector space over the field of real numbers

c)

cannot be determined

d)

$V\ =\left\{\left(x,\ y\right)\ :\ x\ ,\ y\ are\ real\ numbers\right\}$

60.

Let u = (2, 0, −1), v = (3, 1, 0), and w = (1, −1, c) where c ∈ R.

The set {u, v, w} is a basis for $R^3$ provided that c is not equal to

a)

0

b)

2

c)

-2

d)

1

61.

What is the dimension of the space of all matrices A in $R^{2\times2}$ such that $a+d=0$

a)

dim (A) = 3

b)

dim (A) = 4

c)

dim (A) = 2

d)

dim (A) = 0

62.

Which of the following denotes commutative property in vector spaces?

a)

a+b=a-b

b)

a+b=b+a

c)

ab=a

d)

a+b=a+2b

63.

Every set of (n+1) or more vectors in an 'n' dimensional vector space is Linearly independent.

a)

TRUE

b)

FALSE

64.

C(R) is a Vector space

a)

TRUE

b)

FALSE

65.

R(C) is a Vector space

a)

TRUE

b)

FALSE

66.

Intersection of two subspaces is a subspace of vector space

a)

TRUE

b)

FALSE

67.

Union of two subspaces is a subspace of a vector space

a)

TRUE

b)

FALSE

68.

The Sum of two subspaces is a subspace of a vector space

a)

TRUE

b)

FALSE

69.

The set W of ordered triads (x,y,0), where x,y are real numbers, is a subspace of $V_{3\ }\left(R\right)$ .

a)

TRUE

b)

FALSE

70.

W={(a,b,c) : a2+b2+c2<1} is a subspace of V₃(R).

a)

TRUE

b)

FALSE

71.

The set {(1,2,0),(0,3,1),(-1,0,1)} of V₃(Q) is Linearly independent

a)

TRUE

b)

FALSE

72.

A nonempty subset of a Linearly independent set of vectors is Linearly independent.

a)

TRUE

b)

FALSE

73.

A set of vectors, which contains zero vector is Linearly independent.

a)

TRUE

b)

FALSE

74.

If V(F) is a finite-dimensional vector space then there exists a basis set of V.

a)

TRUE

b)

FALSE

75.

If W is a subspace of an 'n' dimensional vector space then dim W >n

a)

TRUE

b)

FALSE

76.

If S is a Linearly independent set of V(F) then S can be extended to form a basis for V.

a)

TRUE

b)

FALSE

77.

Let S be a set of an 'n' dimensional vector space such that L(S)=V then S is a basis of V.

a)

TRUE

b)

FALSE

78.

The set of all linear combinations of given vectors is called the (a) of the set.

79.

The basis for a vector space is ............................................

a)

linearly independent set

b)

spanning set

c)

both linearly independent and spanning set

d)

neither spanning set nor linearly independent

80.

The number of elements in any basis is called (a) of the vector space

81.

The trivial subspaces of V are..........................................

a)

{0}

b)

V

c)

both {0} and V

d)

none of these

82.

which of the following is not a subspace of

R^3

a)

$W=\left\{\left(x,y,z\right)\ |\ \ x+2y=0\right\}$

b)

$W=\left\{\left(x,y,z\right)\ |\ \ x+y=0\right\}$

c)

$W=\left\{\left(x,y,z\right)\ |\ \ x+y+z=0\right\}$

d)

$W=\left\{\left(x,y,z\right)\ |\ \ x+y=1\right\}$

83.

Which of the following is a vector space?

a)

The set of all pairs of real numbers of the form $\left(x,0\right)$ with standard operations on $R^2$

b)

The set of all pairs of real numbers of the form $\left(x,y\right)$ , where $\ x\ge0,$ with the standard operations on $R^2$

c)

The set of all pairs of real numbers with the standard vector addition but with scalar multiplication defined by

$k\left(x,y\right)=\left(k^2x,\ k^2y\right)$

d)

None of the above.

84.

Which of these is a subspace of $M_{nn}$ ?

a)

The set of non-invertible $n\ \times n$ matrices

b)

The set of all $n\times n$ matrices $A$ such that $\det\left(A\right)=0$

c)

The set of all $n\times n$ matrices A such that $tr\left(A\right)=0$

d)

None of these

85.

For a set of vectors to be a vector space, it should satisfy 10 axioms. Which of the following is NOT one of them?

a)

closure under vector addition

b)

closure under scalar multiplication

c)

existence of the neutral element

d)

closure under vector multiplication

86.

Is the set of vectors $\left\{(a,b)\ ∈\ R^2\ :\ b=3a+1\right\}$ a vector space?

a)

Yes

b)

No

c)

Cannot be determined.

87.

The set of all 2x2 matrices with determinant equal to zero under the operations of matrix addition and scalar multiplication is not a vector space. Why?

a)

2x2 matrices are not vectors.

b)

With matrices, AB need not equal BA.

c)

d)

e)

None of the above.

88.

$\mathbb{R}^2$ Which of the three sets in the plane is a subspace of
the vector space

a)

The blue line only.

b)

The red line only

c)

The green line only.

d)

More than one.

e)

None of them.

89.

What is a vector space?

a)

Describes a collection of objects called vectors, which can be added together and scaled by numbers.

b)

Describes a collection of objects called vectors, which can be divided together.

c)

Describes as a single number which can be added together.

d)

None of these.

90.

What is the span of a set of vectors in a vector space? A) The set of all possible linear combinations of the vectors B) The set of all possible scalar multiples of the vectors C) The set of all possible dot products of the vectors D) The set of all possible cross products of the vectors

a)

A) The set of all possible linear combinations of the vectors

b)

B) The set of all possible scalar multiples of the vectors

c)

C) The set of all possible dot products of the vectors

d)

D) The set of all possible cross products of the vectors

91.

Which of the following is NOT a vector space property? A) Distributivity of scalar multiplication over vector addition B) Existence of additive inverse C) Existence of multiplicative inverse D) Associativity of scalar multiplication

a)

D) Associativity of scalar multiplication

b)

B) Existence of additive inverse

c)

C) Existence of multiplicative inverse

d)

A) Distributivity of scalar multiplication over vector addition

92.

If two vectors in a vector space are linearly dependent, what can be said about their scalar multiples? A) They are linearly independent B) They are equal C) They are parallel D) They are orthogonal

a)

C) They are parallel

b)

A) They are linearly independent

c)

B) They are equal

d)

D) They are orthogonal

93.

What is the span of a set of vectors in a vector space? A) The set of all possible linear combinations of the vectors B) The set of all possible scalar multiples of the vectors C) The set of all possible dot products of the vectors D) The set of all possible cross products of the vectors

a)

A) The set of all possible linear combinations of the vectors

b)

B) The set of all possible scalar multiples of the vectors

c)

C) The set of all possible dot products of the vectors

d)

D) The set of all possible cross products of the vectors

94.

What does it mean for vectors to be linearly independent?

a)

They all have the same magnitude

b)

They can be expressed as a linear combination of each other

c)

They all point in the same direction

d)

They cannot be expressed as a linear combination of each other

95.

What does the span of a set of vectors represent?

a)

The maximum length of the vectors

b)

The orthogonal complement of the vectors

c)

The set of all linear combinations of those vectors

d)

The average of the vectors

96.

Which of the following is an example of a vector space?

a)

The set of all integers

b)

The set of all 2x3 matrices

c)

The set of all positive real numbers

d)

The set of all complex numbers

97.

A set of linear equations is represented by the matrix equation Ax=b. The necessary condition for the existence of a solution for this system is

a)

A must be invertible

b)

b must be linearly dependent on columns of A

c)

b must be linearly independent of columns of A

d)

None of these

98.

Which of the following would be a linear combination of {(a, 0), (b, 0)} ? (a and b are non-zero)

a)

(3,0)

b)

(1, 3)

c)

(0,5)

d)

(16, 0)

99.

What is the dimension of real polynomial of degree n over the set of reals?

a)

n

b)

n-1

c)

n+1

d)

not defined

Vector Space-Subspace-Linearly Independent-Bases-Dimension

Total questions: 99

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