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Vector Spaces

Description: This quiz is designed to assess your understanding of the fundamental concepts related to vector spaces, including vector operations, linear independence, span, and subspaces.
Number of Questions: 14
Created by:
Tags: vector spaces linear algebra mathematics
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Let (V) be a vector space over a field (F). Which of the following statements is true about the zero vector (\mathbf{0}) in (V)?

  1. The zero vector is unique.

  2. The zero vector is the only vector in (V).

  3. The zero vector is the additive inverse of itself.

  4. The zero vector is the multiplicative inverse of itself.

Show answer
Correct Option: A
Explanation:

In any vector space, there is a unique zero vector that serves as the additive identity. This means that for any vector (\mathbf{v}) in (V), the sum (\mathbf{v} + \mathbf{0} = \mathbf{v}).

Consider the set of all polynomials with real coefficients. Which of the following operations defines a vector space structure on this set?

  1. Vector addition: ((p + q)(x) = p(x) + q(x)) and scalar multiplication: ((\alpha p)(x) = \alpha p(x))

  2. Vector addition: ((p + q)(x) = p(x) - q(x)) and scalar multiplication: ((\alpha p)(x) = \alpha p(x))

  3. Vector addition: ((p + q)(x) = p(x) + q(x)) and scalar multiplication: ((\alpha p)(x) = \alpha p(x) + \beta)

  4. Vector addition: ((p + q)(x) = p(x) - q(x)) and scalar multiplication: ((\alpha p)(x) = \alpha p(x) + \beta)

Show answer
Correct Option: A
Explanation:

For a set of polynomials to form a vector space, it must satisfy the axioms of a vector space, including the associative, commutative, and distributive properties. The given operations satisfy these properties, making them a valid vector space structure.

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true about linear independence?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then (\mathbf{v}_1 + \mathbf{v}_2 + \mathbf{v}_3 = \mathbf{0}).

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then no vector in the set can be expressed as a linear combination of the others.

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they span the entire vector space (V).

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they form a basis for (V).

Show answer
Correct Option: B
Explanation:

Linear independence means that no vector in the set can be written as a linear combination of the other vectors in the set. This implies that none of the vectors can be expressed as a multiple of the others.

Let (V) be a vector space and (S = {\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3} ) be a subset of (V). Which of the following statements is true about the span of (S)?

  1. The span of (S) is the set of all linear combinations of the vectors in (S).

  2. The span of (S) is the smallest subspace of (V) that contains (S).

  3. The span of (S) is the largest subspace of (V) that contains (S).

  4. The span of (S) is the set of all vectors in (V) that are orthogonal to the vectors in (S).

Show answer
Correct Option: A
Explanation:

The span of a set of vectors is the set of all possible linear combinations of those vectors. It is the smallest subspace of the vector space that contains the given set of vectors.

Let (V) be a vector space and (W) be a subspace of (V). Which of the following statements is always true?

  1. Every vector in (W) is also in (V).

  2. Every vector in (V) is also in (W).

  3. The dimension of (W) is always less than or equal to the dimension of (V).

  4. The dimension of (W) is always greater than or equal to the dimension of (V).

Show answer
Correct Option: A
Explanation:

By definition, a subspace is a subset of a vector space that is itself a vector space. Therefore, every vector in a subspace is also in the larger vector space.

Which of the following sets of vectors is linearly independent in (\mathbb{R}^3)?

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

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

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

  4. ({(1, 1, 1), (1, 1, -1), (1, -1, 1)})

Show answer
Correct Option: A
Explanation:

The set ({(1, 0, 0), (0, 1, 0), (0, 0, 1)}) is linearly independent because no vector in the set can be expressed as a linear combination of the other two vectors.

Let (V) be a vector space of dimension (n). Which of the following statements is true?

  1. Any set of (n) linearly independent vectors in (V) forms a basis for (V).

  2. Any set of (n) vectors in (V) forms a basis for (V).

  3. Any set of (n + 1) linearly independent vectors in (V) forms a basis for (V).

  4. Any set of (n - 1) linearly independent vectors in (V) forms a basis for (V).

Show answer
Correct Option: A
Explanation:

A basis for a vector space is a set of linearly independent vectors that span the entire space. Since the dimension of (V) is (n), any set of (n) linearly independent vectors will span the entire space and thus form a basis.

Which of the following sets of vectors is a subspace of (\mathbb{R}^4)?

  1. ({(1, 2, 3, 4), (2, 4, 6, 8), (3, 6, 9, 12)})

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

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

  4. ({(1, 2, 3, 4), (2, 4, 6, 7), (3, 6, 9, 11)})

Show answer
Correct Option: B
Explanation:

A subspace of a vector space must be closed under vector addition and scalar multiplication. The set ({(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}) satisfies these properties and is therefore a subspace of (\mathbb{R}^4).

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then (\mathbf{v}_1 + \mathbf{v}_2 + \mathbf{v}_3 = \mathbf{0}).

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they span the entire vector space (V).

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they form a basis for (V).

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they are orthogonal to each other.

Show answer
Correct Option: C
Explanation:

A basis for a vector space is a set of linearly independent vectors that span the entire space. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they cannot be expressed as linear combinations of each other. Additionally, if they span the entire space, then any vector in (V) can be expressed as a linear combination of (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3). Therefore, they form a basis for (V).

Let (V) be a vector space and (W) be a subspace of (V). Which of the following statements is true?

  1. The intersection of (V) and (W) is always a subspace of (V).

  2. The union of (V) and (W) is always a subspace of (V).

  3. The complement of (W) in (V) is always a subspace of (V).

  4. The direct sum of (V) and (W) is always a subspace of (V).

Show answer
Correct Option: A
Explanation:

The intersection of two subspaces is also a subspace because it inherits the properties of being closed under vector addition and scalar multiplication from both subspaces.

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly dependent, then they span the entire vector space (V).

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly dependent, then they form a basis for (V).

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly dependent, then they are orthogonal to each other.

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly dependent, then at least one of them can be expressed as a linear combination of the others.

Show answer
Correct Option: D
Explanation:

Linear dependence means that at least one of the vectors can be expressed as a linear combination of the others. This implies that the vectors are not linearly independent.

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are orthogonal to each other, then they are linearly independent.

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are orthogonal to each other, then they span the entire vector space (V).

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are orthogonal to each other, then they form a basis for (V).

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are orthogonal to each other, then they are linearly dependent.

Show answer
Correct Option: A
Explanation:

Orthogonal vectors are linearly independent because none of them can be expressed as a linear combination of the others. This is because the dot product of any two orthogonal vectors is zero.

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) span the entire vector space (V), then they are linearly independent.

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) span the entire vector space (V), then they form a basis for (V).

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) span the entire vector space (V), then they are orthogonal to each other.

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) span the entire vector space (V), then they are linearly dependent.

Show answer
Correct Option: B
Explanation:

A basis for a vector space is a set of vectors that span the entire space and are linearly independent. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) span the entire space, then they can be used to express any vector in (V) as a linear combination. Additionally, if they are linearly independent, then they cannot be expressed as linear combinations of each other. Therefore, they form a basis for (V).

Let (V) be a vector space and (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) be vectors in (V). Which of the following statements is true?

  1. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they are orthogonal to each other.

  2. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they span the entire vector space (V).

  3. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they form a basis for (V).

  4. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they are linearly dependent.

Show answer
Correct Option: C
Explanation:

A basis for a vector space is a set of vectors that span the entire space and are linearly independent. If (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3) are linearly independent, then they cannot be expressed as linear combinations of each other. Additionally, if they span the entire space, then any vector in (V) can be expressed as a linear combination of (\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3). Therefore, they form a basis for (V).

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