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.

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(Key: correct, incorrect, partially correct.)

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

Score:  

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