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

(Key: correct, incorrect, partially correct.)

- Let be an vector space, and let be a subset of . What does it mean when we say that is closed under addition?
- Whenever and are in , then is in .
- Whenever and are in , then is in .
- Whenever and are in , then is in .
- CORRECT.
- Whenever and are in , then is in .
- If is in , then and are in .
- for every two vectors and .
- Every vector in is the sum of two vectors in .
- Let be a vector space, and let be a subset of . What does it mean when we say that is closed under scalar multiplication?
- Whenever is in and is a scalar, then is in .
- Whenever is in and is a scalar, then is in .
- Whenever is in and is a scalar, then is in .
- CORRECT.
- Whenever is in and is a scalar, then is in .
- If is in and is in , then is a scalar.
- If is in and is a scalar, then is in .
- PARTIALLY.
- for every vector and scalar .
- Let be a vector space, and let be a subset of . What does it mean when we say that is linearly independent?
- is closed under both addition and scalar multiplication.
- is a basis.
- INCORRECT. Every basis is linearly independent, but not vice versa.
- Every element in is a linear combination of elements in .
- 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.
- 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.
- All the elements of are distinct from each other.
- has nullity zero.
- Let be a vector space, and let be a subset of . What does it mean when we say that is linearly dependent?
- is closed under both addition and scalar multiplication.
- 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.
- The number of elements of is greater than the dimension of .
- PARTIALLY.
- There is a way to write as a linear combination of elements of other than the zero combination.
- CORRECT.
- The span of has smaller dimension than the dimension of .
- depends on a linear transformation.
- At least two of the elements of are the same.
- Let be a vector space, and let be a subset of . What does it mean when we say that spans ?
- is a basis for .
- The elements of are all distinct from each other.
- Every vector in can be expressed as a linear combination of vectors in .
- CORRECT.
- Every vector in has exactly one representation as a linear combination of vectors in .
- 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.
- 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.
- Let be a five-dimensional vector space, and let be a subset of which spans . Then
- Must consist of at least five elements.
- CORRECT.
- Must have exactly five elements.
- Must have at most five elements.
- Must have infinitely many elements.
- Must be linearly independent.
- Must be a basis for .
- Must be linearly dependent.
- Let be a five-dimensional vector space, and let be a subset of which is linearly independent. Then
- Must consist of at least five elements.
- Must have exactly five elements.
- Must have at most five elements.
- CORRECT.
- Must have infinitely many elements.
- Must span .
- Must be a basis for .
- Can have any number of elements (except zero).
- Let be a five-dimensional vector space, and let be a subset of which is linearly dependent. Then
- Must consist of at least five elements.
- Must have exactly five elements.
- Must have at most five elements.
- Must have infinitely many elements.
- Must span .
- Must be a basis for .
- Can have any number of elements (except zero).
- CORRECT.
- Let be a five-dimensional vector space, and let be a subset of which is a basis for . Then
- Must consist of at least five elements.
- PARTIALLY. While it is true that must contain at least five elements, you can say something similar and stronger.
- Must have exactly five elements.
- CORRECT.
- Must have at most five elements.
- PARTIALLY. While it is true that must contain at most five elements, you can say something similar and stronger.
- Must be linearly dependent.
- Must span .
- CORRECT.
- Must be linearly independent.
- CORRECT.
- Can have any number of elements (except zero).
- Let be a five-dimensional vector space, and let be a subset of consisting of three vectors. Then
- Cannot span , but can be linearly independent or dependent.
- CORRECT.
- Must be linearly independent, but may or may not span .
- Must be linearly dependent, and must span .
- May or may not be linearly independent, and may or may not span .
- Must be linearly dependent, but may or may not span .
- Must be linearly independent, but cannot span .
- Can span , but only if it is linearly independent, and vice versa.
- Let be a three-dimensional vector space, and let be a subset of consisting of five vectors. Then
- Cannot span , but can be linearly independent or dependent.
- Must be linearly dependent, and must span .
- Must be linearly independent, but may or may not span .
- May or may not be linearly independent, and may or may not span .
- Must be linearly dependent, but may or may not span .
- CORRECT.
- Must be linearly independent, but cannot span .
- Can span , but only if it is linearly independent, and vice versa.
- Let be a five-dimensional vector space, and let be a subset of consisting of five vectors. Then
- Cannot span , but can be linearly independent or dependent.
- Must be linearly dependent, and must span .
- Must be linearly independent, but may or may not span .
- Must be a basis of .
- Must be linearly dependent, but may or may not span .
- Must be linearly independent, but cannot span .
- Can span , but only if it is linearly independent, and vice versa.
- CORRECT.
- If are five vectors in , then the number of redundant vectors
- Can be any number from two to five.
- CORRECT.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- Is three.
- If are three vectors in , then the number of redundant vectors
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.
- If are five vectors in , then the number of non-redundant vectors
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.
- If are three vectors in , then the number of non-redundant vectors
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.
- The rank of a matrix
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.
- The nullity of a matrix
- Can be any number from two to five.
- CORRECT.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- Is three.
- The rank of a matrix
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.
- The nullity of a matrix
- Can be any number from two to five.
- Must be two.
- Can be any number from zero to two.
- Can be any number from zero to five.
- Must be zero.
- Can be any number from zero to three.
- CORRECT.
- Is three.

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