Vector Spaces: A Comprehensive Guide for Engineering Mathematics

Question 1

Consider the set V = {a + bx + cx² | a, b, c ∈ R}. Determine whether V is a vector space over R under the usual operations of vector addition and scalar multiplication.

Accepted Answer

V is a vector space, it satisfies the vector space axioms.

Answer

V is not a vector space, it does not contain the zero vector.

Answer

V is not a vector space, scalar multiplication is not defined for all elements of V.

Answer

V is not a vector space, it is not closed under vector addition.

Question 2

Determine whether the set of vectors {(1, 2, 3), (2, 4, 6), (3, 6, 9)} in R³ is linearly independent.

Accepted Answer

The set is linearly independent.

Answer

The set is linearly dependent because (1, 2, 3) = (2, 4, 6)/2.

Answer

The set is linearly dependent because (2, 4, 6) = 2(1, 2, 3).

Answer

The set is linearly dependent because (3, 6, 9) = 3(1, 2, 3).

Question 3

Find a basis for the vector space V = {a + bx | a, b ∈ R}.

Accepted Answer

{1, x}

Answer

{1, x, x²}

Answer

{1, x, x², x³}

Answer

{1, x, x², x³, x⁴}

Question 4

Which of the following is a defining property of a vector space?

Accepted Answer

Closure under vector addition and scalar multiplication

Answer

Existence of a unique zero vector

Answer

Associative property for vector addition

Answer

Distributive property for scalar multiplication over vector addition

Question 5

A set of vectors {v1, v2} is linearly independent if and only if:

Accepted Answer

cv1 + dv2 = 0 only if c = d = 0

Answer

v1 + v2 = 0

Answer

v1 is a non-zero multiple of v2

Answer

v2 is a non-zero multiple of v1

Question 6

Which of the following sets of vectors spans the two-dimensional vector space R^2?

Accepted Answer

{[1, 0], [0, 1]}

Answer

{[1, 1], [1, -1]}

Answer

{[1, 0], [1, 1]}

Answer

{[0, 1], [1, 0]}

Question 7

Consider a vector space V with vectors u and v. If u is linearly dependent on v, then:

Accepted Answer

u = 0

Answer

v = 0

Answer

u and v are linearly independent

Answer

u and v span V

Question 8

Which of the following matrices represents a linear transformation from R^2 to R^3?

Accepted Answer

2 x 3

Answer

3 x 2

Answer

3 x 3

Answer

2 x 2

Question 9

Which of the following sets is not a subspace of the vector space R^3?

Accepted Answer

Set D: Vectors with zero sum of components

Answer

Set A: Vectors with zero first component

Answer

Set B: Vectors with zero second component

Answer

Set C: Vectors with zero third component

Question 10

Consider the set of all polynomials of degree less than or equal to 2. Which of the following is a basis for this vector space?

Accepted Answer

{1, x, x^2}

Answer

{1, x, x^3}

Answer

{1, x^2, x^3}

Answer

{1, x, x^2, x^3}

Question 11

Determine whether the set of vectors {(1, 2), (3, 4), (5, 6)} is linearly independent in R^2.

Accepted Answer

Linearly independent

Answer

Linearly dependent

Question 12

Which of the following statements is equivalent to the statement that a set of vectors spans a vector space?

Accepted Answer

Every vector in the space can be expressed as a linear combination of the vectors in the set

Answer

The set is linearly independent

Answer

The set is a basis for the space

Answer

The set contains the zero vector

Question 13

The vector space of all real square matrices of order 2 has dimension:

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 14

Consider the set of all polynomials with real coefficients of degree less than or equal to 3. Is this set a vector space under the usual operations of polynomial addition and scalar multiplication?

Accepted Answer

Yes

Answer

No

Question 15

Determine if the following set of vectors is linearly independent: {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Accepted Answer

Linearly independent

Answer

Linearly dependent

Question 16

If a vector space has a spanning set of n vectors, what is the maximum possible number of linearly independent vectors it can have?

Accepted Answer

n

Answer

n+1

Answer

n-1

Answer

2n

Question 17

Let V be a vector space over a field F. Which of the following is NOT a subspace of V?

Accepted Answer

The set of all vectors that are orthogonal to a given vector

Answer

The set of all vectors that are scalar multiples of a given vector

Answer

The set of all zero vectors

Answer

The set of all vectors that satisfy a given linear equation

Question 18

Which of the following is NOT a property of a vector space?

Accepted Answer

Existence of an identity vector

Answer

Closure under addition and scalar multiplication

Answer

Associativity of addition

Answer

Existence of a zero vector

Question 19

Consider the set of all polynomials with real coefficients. Is this set a vector space under the usual operations of polynomial addition and scalar multiplication?

Accepted Answer

Yes, it satisfies all the properties of a vector space

Answer

No, it does not have an identity vector

Answer

No, it is not closed under scalar multiplication

Answer

No, it is not closed under addition

Question 20

Which of the following is NOT a defining property of a vector space?

Accepted Answer

Non-associative operation

Answer

Existence of additive inverse

Answer

Closure under addition

Answer

Existence of zero vector

Question 21

Which of the following sets of vectors forms a spanning set for the vector space R^n?

Accepted Answer

{(1, 0, ..., 0), (0, 1, ..., 0), ..., (0, ..., 0, 1)}

Answer

{(1, 1, ..., 1), (1, 0, ..., 0), ..., (0, ..., 0, 1)}

Answer

{(1, 0, ..., 0), (0, 1, ..., 0), ..., (0, ..., 0, 2)}

Answer

{(1, 2, ..., n), (2, 3, ..., n+1), ..., (n, n+1, ..., 2n)}

Question 22

Let V be a finite-dimensional vector space with basis B = {v1, v2, ..., vn}. If x is a vector in V, its unique representation in terms of B is:

Accepted Answer

x = a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars

Answer

x = v1 + v2 + ... + vn

Answer

x is not uniquely representable.

Answer

x = a1 + a2 + ... + an, where a1, a2, ..., an are scalars

Question 23

Consider the vector space of real functions continuous on the interval [0, 1]. Which of the following functionals is linear?

Accepted Answer

∫[0,1] f(x) dx

Answer

∫[0,1] f(x) * sin(x) dx

Answer

max{|f(x)|} on [0, 1]

Answer

f(0)

Question 24

Let V be an inner product space and let u and v be vectors in V. The Cauchy-Schwarz inequality states that:

Accepted Answer

|<u, v>| ≤ ||u|| ||v||

Answer

||u - v|| ≥ ||u|| - ||v||

Answer

<u, v> = 0 if and only if u = v

Answer

||u + v|| ≤ ||u|| + ||v||

Question 25

Let A be an n x n matrix. The null space of A is the set of all vectors x such that:

Accepted Answer

Ax = 0

Answer

x is an eigenvector of A

Answer

x^T A = 0

Answer

Ax = b, where b is a non-zero vector