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Vector Spaces and Subspaces

This quiz is designed to assess your understanding of vector spaces and subspaces, including concepts like linear combinations, span, and independence.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

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
Question 2 Multiple Choice (Single Answer)

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
Question 3 Multiple Choice (Single Answer)

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
Question 4 Multiple Choice (Single Answer)

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
Question 5 Multiple Choice (Single Answer)

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
Question 6 Multiple Choice (Single Answer)

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
Question 7 Multiple Choice (Single Answer)

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
Question 8 Multiple Choice (Single Answer)

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
Question 9 Multiple Choice (Single Answer)

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
Question 10 Multiple Choice (Single Answer)

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
Question 11 Multiple Choice (Single Answer)

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
Question 12 Multiple Choice (Single Answer)

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
Question 13 Multiple Choice (Single Answer)

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
Question 14 Multiple Choice (Single Answer)

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
Question 15 Multiple Choice (Single Answer)

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
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