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Basis and Dimension

This quiz is designed to assess your understanding of the concepts related to basis and dimension in linear algebra.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

In a vector space, a set of vectors is said to be linearly independent if:

  1. Every vector in the set can be expressed as a linear combination of the other vectors in the set.
  2. No vector in the set can be expressed as a linear combination of the other vectors in the set.
  3. The set contains the zero vector.
  4. The set contains more vectors than the dimension of the vector space.
Question 2 Multiple Choice (Single Answer)

The dimension of a vector space is:

  1. The number of vectors in a basis for the vector space.
  2. The number of linearly independent vectors in the vector space.
  3. The number of linearly dependent vectors in the vector space.
  4. The number of vectors that span the vector space.
Question 3 Multiple Choice (Single Answer)

Which of the following sets of vectors is a basis for R^3?

  1. {(1, 0, 0), (0, 1, 0), (0, 0, 1)}
  2. {(1, 1, 1), (1, 1, 0), (1, 0, 1)}
  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}
  4. {(1, 0, 0), (0, 1, 1), (1, 1, 0)}
Question 4 Multiple Choice (Single Answer)

If a vector space has a finite basis, then it is called:

  1. Finite-dimensional vector space
  2. Infinite-dimensional vector space
  3. Linearly independent vector space
  4. Spanning vector space
Question 5 Multiple Choice (Single Answer)

The dimension of the vector space of all polynomials of degree less than or equal to n is:

  1. n
  2. n+1
  3. n-1
  4. 2n
Question 6 Multiple Choice (Single Answer)

Which of the following sets of vectors is linearly dependent?

  1. {(1, 0, 0), (0, 1, 0), (0, 0, 1)}
  2. {(1, 1, 1), (1, 1, 0), (1, 0, 1)}
  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}
  4. {(1, 0, 0), (0, 1, 1), (1, 1, 0)}
Question 7 Multiple Choice (Single Answer)

If a set of vectors spans a vector space, then it is called:

  1. A basis for the vector space
  2. A linearly independent set
  3. A linearly dependent set
  4. A subspace of the vector space
Question 8 Multiple Choice (Single Answer)

The dimension of the vector space of all real-valued functions that are continuous on the interval [0, 1] is:

  1. Infinite
  2. 1
  3. 2
  4. 3
Question 9 Multiple Choice (Single Answer)

Which of the following sets of vectors is a basis for the vector space of all polynomials of degree less than or equal to 2?

  1. {(1, 0, 0), (0, 1, 0), (0, 0, 1)}
  2. {(1, 1, 1), (1, 1, 0), (1, 0, 1)}
  3. {(1, 2, 3), (4, 5, 6), (7, 8, 9)}
  4. {(1, 0, 0), (0, 1, 1), (1, 1, 0)}
Question 10 Multiple Choice (Single Answer)

If a set of vectors is linearly independent, then it is called:

  1. A basis for the vector space
  2. A linearly dependent set
  3. A spanning set for the vector space
  4. A subspace of the vector space
Question 11 Multiple Choice (Single Answer)

The dimension of the vector space of all real-valued functions that are differentiable on the interval [0, 1] is:

  1. Infinite
  2. 1
  3. 2
  4. 3
Question 12 Multiple Choice (Single Answer)

Which of the following sets of vectors is a basis for the vector space of all polynomials of degree less than or equal to 3?

  1. {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}
  2. {(1, 1, 1, 1), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)}
  3. {(1, 2, 3, 4), (4, 5, 6, 7), (7, 8, 9, 10)}
  4. {(1, 0, 0, 0), (0, 1, 1, 0), (1, 1, 0, 1), (1, 1, 1, 0)}
Question 13 Multiple Choice (Single Answer)

If a vector space has an infinite basis, then it is called:

  1. Finite-dimensional vector space
  2. Infinite-dimensional vector space
  3. Linearly independent vector space
  4. Spanning vector space
Question 14 Multiple Choice (Single Answer)

The dimension of the vector space of all real-valued functions that are continuous on the interval [0, ∞) is:

  1. Infinite
  2. 1
  3. 2
  4. 3
Question 15 Multiple Choice (Single Answer)

Which of the following sets of vectors is a basis for the vector space of all polynomials of degree less than or equal to 4?

  1. {(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)}
  2. {(1, 1, 1, 1, 1), (1, 1, 0, 0, 0), (1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (1, 0, 0, 0, 1)}
  3. {(1, 2, 3, 4, 5), (4, 5, 6, 7, 8), (7, 8, 9, 10, 11)}
  4. {(1, 0, 0, 0, 0), (0, 1, 1, 0, 0), (1, 1, 0, 1, 0), (1, 1, 1, 0, 1), (1, 1, 1, 1, 0)}
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