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Linear Independence

Description: This quiz is designed to assess your understanding of the concept of linear independence in linear algebra.
Number of Questions: 14
Created by:
Tags: linear algebra linear independence vector spaces
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Which of the following sets of vectors is linearly independent in the vector space of polynomials of degree 2 or less?

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

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

  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

  4. {(1, 0, 1), (0, 1, 1), (1, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(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 vectors.

Determine whether the following set of vectors is linearly independent in the vector space of real numbers:

  1. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

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

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

  4. {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2, 3), (4, 5, 6), (7, 8, 9)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of continuous functions on the interval [0, 1]:

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

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

  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

  4. {(1, 0, 1), (0, 1, 1), (1, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(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 vectors.

Determine whether the following set of vectors is linearly independent in the vector space of complex numbers:

  1. {(1, 2i), (3, 4i), (5, 6i)}

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

  3. {(1, 2), (3, 4), (5, 6)}

  4. {(1, 1), (1, 0), (0, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2i), (3, 4i), (5, 6i)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

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

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

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}

Show answer
Correct Option: A
Explanation:

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

Determine whether the following set of vectors is linearly independent in the vector space of real numbers:

  1. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

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

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

  4. {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2, 3), (4, 5, 6), (7, 8, 9)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of continuous functions on the interval [0, 1]:

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

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

  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

  4. {(1, 0, 1), (0, 1, 1), (1, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(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 vectors.

Determine whether the following set of vectors is linearly independent in the vector space of complex numbers:

  1. {(1, 2i), (3, 4i), (5, 6i)}

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

  3. {(1, 2), (3, 4), (5, 6)}

  4. {(1, 1), (1, 0), (0, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2i), (3, 4i), (5, 6i)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

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

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

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}

Show answer
Correct Option: A
Explanation:

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

Determine whether the following set of vectors is linearly independent in the vector space of real numbers:

  1. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

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

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

  4. {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2, 3), (4, 5, 6), (7, 8, 9)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of continuous functions on the interval [0, 1]:

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

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

  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

  4. {(1, 0, 1), (0, 1, 1), (1, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(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 vectors.

Determine whether the following set of vectors is linearly independent in the vector space of complex numbers:

  1. {(1, 2i), (3, 4i), (5, 6i)}

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

  3. {(1, 2), (3, 4), (5, 6)}

  4. {(1, 1), (1, 0), (0, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2i), (3, 4i), (5, 6i)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

Consider the following set of vectors in the vector space of polynomials of degree 3 or less:

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

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

  3. {(1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12)}

  4. {(1, 0, 1, 1), (0, 1, 1, 1), (1, 1, 1, 1)}

Show answer
Correct Option: A
Explanation:

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

Determine whether the following set of vectors is linearly independent in the vector space of real numbers:

  1. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}

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

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

  4. {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

Show answer
Correct Option: A
Explanation:

The set of vectors {(1, 2, 3), (4, 5, 6), (7, 8, 9)} is linearly independent because no vector in the set can be expressed as a linear combination of the other vectors.

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