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Vector Spaces and Subspaces

Description: This quiz is designed to assess your understanding of vector spaces and subspaces, including concepts like linear combinations, span, and independence.
Number of Questions: 15
Created by:
Tags: linear algebra vector spaces subspaces linear combinations span independence
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Which of the following is a vector space over the field of real numbers?

  1. The set of all 2x2 matrices with real entries

  2. The set of all polynomials of degree at most 3

  3. The set of all functions from the real numbers to the real numbers

  4. The set of all ordered pairs of real numbers

Show answer
Correct Option: D
Explanation:

A vector space over a field F is a non-empty set V together with two operations, vector addition and scalar multiplication, that satisfy certain axioms. The set of all ordered pairs of real numbers, denoted by R^2, forms a vector space over the field of real numbers because it satisfies all the axioms of a vector space.

Let V be a vector space over a field F. Which of the following is a subspace of V?

  1. The set of all vectors in V that have a zero first component

  2. The set of all vectors in V that have a non-zero first component

  3. The set of all vectors in V that have a zero last component

  4. The set of all vectors in V that have a non-zero last component

Show answer
Correct Option: A
Explanation:

A subspace of a vector space V over a field F is a non-empty subset W of V that is itself a vector space over F. The set of all vectors in V that have a zero first component forms a subspace of V because it is non-empty, closed under vector addition, and closed under scalar multiplication.

Let V be a vector space over a field F. Which of the following is a linear combination of the vectors v1, v2, ..., vn in V?

  1. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F

  2. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are vectors in V

  3. v1 + v2 + ... + vn

  4. v1 - v2 + ... - vn

Show answer
Correct Option: A
Explanation:

A linear combination of the vectors v1, v2, ..., vn in a vector space V over a field F is a vector of the form a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F.

Let V be a vector space over a field F. Which of the following is a span of the vectors v1, v2, ..., vn in V?

  1. The set of all linear combinations of v1, v2, ..., vn

  2. The set of all vectors in V that can be expressed as a linear combination of v1, v2, ..., vn

  3. The set of all vectors in V that are equal to v1, v2, ..., vn

  4. The set of all vectors in V that are not equal to v1, v2, ..., vn

Show answer
Correct Option: A
Explanation:

The span of the vectors v1, v2, ..., vn in a vector space V over a field F is the set of all linear combinations of v1, v2, ..., vn.

Let V be a vector space over a field F. Which of the following is a linearly independent set of vectors in V?

  1. A set of vectors that spans V

  2. A set of vectors that is not a basis for V

  3. A set of vectors that is not a subspace of V

  4. A set of vectors that is not linearly dependent

Show answer
Correct Option: D
Explanation:

A linearly independent set of vectors in a vector space V over a field F is a set of vectors that is not linearly dependent.

Let V be a vector space over a field F. Which of the following is a basis for V?

  1. A linearly independent set of vectors that spans V

  2. A linearly dependent set of vectors that spans V

  3. A linearly independent set of vectors that does not span V

  4. A linearly dependent set of vectors that does not span V

Show answer
Correct Option: A
Explanation:

A basis for a vector space V over a field F is a linearly independent set of vectors that spans V.

Let V be a vector space over a field F. Which of the following is the dimension of V?

  1. The number of vectors in a basis for V

  2. The number of vectors in a linearly independent set of vectors in V

  3. The number of vectors in a spanning set of vectors for V

  4. The number of vectors in a linearly dependent set of vectors in V

Show answer
Correct Option: A
Explanation:

The dimension of a vector space V over a field F is the number of vectors in a basis for V.

Let V be a vector space over a field F. Which of the following is a subspace of V?

  1. The set of all vectors in V that have a zero first component

  2. The set of all vectors in V that have a non-zero first component

  3. The set of all vectors in V that have a zero last component

  4. The set of all vectors in V that have a non-zero last component

Show answer
Correct Option: A
Explanation:

A subspace of a vector space V over a field F is a non-empty subset W of V that is itself a vector space over F. The set of all vectors in V that have a zero first component forms a subspace of V because it is non-empty, closed under vector addition, and closed under scalar multiplication.

Let V be a vector space over a field F. Which of the following is a linear combination of the vectors v1, v2, ..., vn in V?

  1. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F

  2. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are vectors in V

  3. v1 + v2 + ... + vn

  4. v1 - v2 + ... - vn

Show answer
Correct Option: A
Explanation:

A linear combination of the vectors v1, v2, ..., vn in a vector space V over a field F is a vector of the form a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F.

Let V be a vector space over a field F. Which of the following is a span of the vectors v1, v2, ..., vn in V?

  1. The set of all linear combinations of v1, v2, ..., vn

  2. The set of all vectors in V that can be expressed as a linear combination of v1, v2, ..., vn

  3. The set of all vectors in V that are equal to v1, v2, ..., vn

  4. The set of all vectors in V that are not equal to v1, v2, ..., vn

Show answer
Correct Option: A
Explanation:

The span of the vectors v1, v2, ..., vn in a vector space V over a field F is the set of all linear combinations of v1, v2, ..., vn.

Let V be a vector space over a field F. Which of the following is a linearly independent set of vectors in V?

  1. A set of vectors that spans V

  2. A set of vectors that is not a basis for V

  3. A set of vectors that is not a subspace of V

  4. A set of vectors that is not linearly dependent

Show answer
Correct Option: D
Explanation:

A linearly independent set of vectors in a vector space V over a field F is a set of vectors that is not linearly dependent.

Let V be a vector space over a field F. Which of the following is a basis for V?

  1. A linearly independent set of vectors that spans V

  2. A linearly dependent set of vectors that spans V

  3. A linearly independent set of vectors that does not span V

  4. A linearly dependent set of vectors that does not span V

Show answer
Correct Option: A
Explanation:

A basis for a vector space V over a field F is a linearly independent set of vectors that spans V.

Let V be a vector space over a field F. Which of the following is the dimension of V?

  1. The number of vectors in a basis for V

  2. The number of vectors in a linearly independent set of vectors in V

  3. The number of vectors in a spanning set of vectors for V

  4. The number of vectors in a linearly dependent set of vectors in V

Show answer
Correct Option: A
Explanation:

The dimension of a vector space V over a field F is the number of vectors in a basis for V.

Let V be a vector space over a field F. Which of the following is a subspace of V?

  1. The set of all vectors in V that have a zero first component

  2. The set of all vectors in V that have a non-zero first component

  3. The set of all vectors in V that have a zero last component

  4. The set of all vectors in V that have a non-zero last component

Show answer
Correct Option: A
Explanation:

A subspace of a vector space V over a field F is a non-empty subset W of V that is itself a vector space over F. The set of all vectors in V that have a zero first component forms a subspace of V because it is non-empty, closed under vector addition, and closed under scalar multiplication.

Let V be a vector space over a field F. Which of the following is a linear combination of the vectors v1, v2, ..., vn in V?

  1. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F

  2. a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are vectors in V

  3. v1 + v2 + ... + vn

  4. v1 - v2 + ... - vn

Show answer
Correct Option: A
Explanation:

A linear combination of the vectors v1, v2, ..., vn in a vector space V over a field F is a vector of the form a1v1 + a2v2 + ... + anvn, where a1, a2, ..., an are scalars in F.

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