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Quiz: Vector spaces

Page history last edited by WJ 8 years, 5 months ago Saved with comment

TThis quiz is designed to test your knowledge of vector spaces and related concepts such as linear combinations, bases, dimension, spanning, and linear dependence and independence.

 

For an extra challenge, try covering up the answers before attempting the question.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula be an vector space, and let Formula be a subset of Formula. What does it mean when we say that Formula is closed under addition?
    1. Whenever Formula and Formula are in Formula, then Formula is in Formula.
    2. Whenever Formula and Formula are in Formula, then Formula is in Formula.
    3. Whenever Formula and Formula are in Formula, then Formula is in Formula.
      • CORRECT.
    4. Whenever Formula and Formula are in Formula, then Formula is in Formula.
    5. If Formula is in Formula, then Formula and Formula are in Formula.
    6. Formula for every two vectors Formula and Formula.
    7. Every vector in Formula is the sum of two vectors in Formula.
  2. Let Formula be a vector space, and let Formula be a subset of Formula. What does it mean when we say that Formula is closed under scalar multiplication?
    1. Whenever Formula is in Formula and Formula is a scalar, then Formula is in Formula.
    2. Whenever Formula is in Formula and Formula is a scalar, then Formula is in Formula.
    3. Whenever Formula is in Formula and Formula is a scalar, then Formula is in Formula.
      • CORRECT.
    4. Whenever Formula is in Formula and Formula is a scalar, then Formula is in Formula.
    5. If Formula is in Formula and Formula is in Formula, then Formula is a scalar.
    6. If Formula is in Formula and Formula is a scalar, then Formula is in Formula.
      • PARTIALLY.
    7. Formula for every vector Formula and scalar Formula.
  3. Let Formula be a vector space, and let Formula be a subset of Formula. What does it mean when we say that Formula is linearly independent?
    1. Formula is closed under both addition and scalar multiplication.
    2. Formula is a basis.
      • INCORRECT. Every basis is linearly independent, but not vice versa.
    3. Every element in Formula is a linear combination of elements in Formula.
    4. The number of elements of Formula is less than or equal to the dimension of Formula.
      • PARTIALLY. It is true that if Formula is linearly independent, then the number of elements in Formula is less than the dimension of Formula, but it is possible for Formula to have fewer elements than the dimension of Formula without Formula being linearly independent.
    5. The only way to write Formula as a linear combination of elements of Formula is the zero combination (where one takes zero multiples of each element of Formula).
      • CORRECT.
    6. All the elements of Formula are distinct from each other.
    7. Formula has nullity zero.
  4. Let Formula be a vector space, and let Formula be a subset of Formula. What does it mean when we say that Formula is linearly dependent?
    1. Formula is closed under both addition and scalar multiplication.
    2. Every element of Formula is a linear combination of other elements of Formula.
      • INCORRECT. Only one of the elements of Formula needs to be able to be expressed as a combination of the others in order to establish linear dependence.
    3. The number of elements of Formula is greater than the dimension of Formula.
      • PARTIALLY.
    4. There is a way to write Formula as a linear combination of elements of Formula other than the zero combination.
      • CORRECT.
    5. The span of Formula has smaller dimension than the dimension of Formula.
    6. Formula depends on a linear transformation.
    7. At least two of the elements of Formula are the same.
  5. Let Formula be a vector space, and let Formula be a subset of Formula. What does it mean when we say that Formula spans Formula?
    1. Formula is a basis for Formula.
    2. The elements of Formula are all distinct from each other.
    3. Every vector in Formula can be expressed as a linear combination of vectors in Formula.
      • CORRECT.
    4. Every vector in Formula has exactly one representation as a linear combination of vectors in Formula.
    5. Formula has at least as many elements as the dimension of Formula.
      • PARTIALLY. It is necessary for Formula to have at least as many elements as the dimension of Formula in order for Formula to span Formula, but it is not sufficient.
    6. The rank of Formula is the same as the dimension of Formula.
      • PARTIALLY. This is true for finite-dimensional vector spaces, but not for infinite-dimensional ones.
  6. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula which spans Formula. Then Formula
    1. Must consist of at least five elements.
      • CORRECT.
    2. Must have exactly five elements.
    3. Must have at most five elements.
    4. Must have infinitely many elements.
    5. Must be linearly independent.
    6. Must be a basis for Formula.
    7. Must be linearly dependent.
  7. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula which is linearly independent. Then Formula
    1. Must consist of at least five elements.
    2. Must have exactly five elements.
    3. Must have at most five elements.
      • CORRECT.
    4. Must have infinitely many elements.
    5. Must span Formula.
    6. Must be a basis for Formula.
    7. Can have any number of elements (except zero).
  8. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula which is linearly dependent. Then Formula
    1. Must consist of at least five elements.
    2. Must have exactly five elements.
    3. Must have at most five elements.
    4. Must have infinitely many elements.
    5. Must span Formula.
    6. Must be a basis for Formula.
    7. Can have any number of elements (except zero).
      • CORRECT.
  9. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula which is a basis for Formula. Then Formula
    1. Must consist of at least five elements.
      • PARTIALLY.  While it is true that Formula must contain at least five elements, you can say something similar and stronger.
    2. Must have exactly five elements.
      • CORRECT.
    3. Must have at most five elements.
      • PARTIALLY.  While it is true that Formula must contain at most five elements, you can say something similar and stronger.
    4. Must be linearly dependent.
    5. Must span Formula.
      • CORRECT.
    6. Must be linearly independent.
      • CORRECT.
    7. Can have any number of elements (except zero).
  10. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula consisting of three vectors. Then Formula
    1. Cannot span Formula, but can be linearly independent or dependent.
      • CORRECT.
    2. Must be linearly independent, but may or may not span Formula.
    3. Must be linearly dependent, and must span Formula.
    4. May or may not be linearly independent, and may or may not span Formula.
    5. Must be linearly dependent, but may or may not span Formula.
    6. Must be linearly independent, but cannot span Formula.
    7. Can span Formula, but only if it is linearly independent, and vice versa.
  11. Let Formula be a three-dimensional vector space, and let Formula be a subset of Formula consisting of five vectors. Then Formula
    1. Cannot span Formula, but can be linearly independent or dependent.
    2. Must be linearly dependent, and must span Formula.
    3. Must be linearly independent, but may or may not span Formula.
    4. May or may not be linearly independent, and may or may not span Formula.
    5. Must be linearly dependent, but may or may not span Formula.
      • CORRECT.
    6. Must be linearly independent, but cannot span Formula.
    7. Can span Formula, but only if it is linearly independent, and vice versa.
  12. Let Formula be a five-dimensional vector space, and let Formula be a subset of Formula consisting of five vectors. Then Formula
    1. Cannot span Formula, but can be linearly independent or dependent.
    2. Must be linearly dependent, and must span Formula.
    3. Must be linearly independent, but may or may not span Formula.
    4. Must be a basis of Formula.
    5. Must be linearly dependent, but may or may not span Formula.
    6. Must be linearly independent, but cannot span Formula.
    7. Can span Formula, but only if it is linearly independent, and vice versa.
      • CORRECT.
  13. If Formula are five vectors in Formula, then the number of redundant vectors
    1. Can be any number from two to five.
      • CORRECT.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
    7. Is three.
  14. If Formula are three vectors in Formula, then the number of redundant vectors
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.
  15. If Formula are five vectors in Formula, then the number of non-redundant vectors
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.
  16. If Formula are three vectors in Formula, then the number of non-redundant vectors
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.
  17. The rank of a Formula matrix
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.
  18. The nullity of a Formula matrix
    1. Can be any number from two to five.
      • CORRECT.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
    7. Is three.
  19. The rank of a Formula matrix
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.
  20. The nullity of a Formula matrix
    1. Can be any number from two to five.
    2. Must be two.
    3. Can be any number from zero to two.
    4. Can be any number from zero to five.
    5. Must be zero.
    6. Can be any number from zero to three.
      • CORRECT.
    7. Is three.

 

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