Comment. Vector spaces quiz
Comment. This quiz is designed to test your knowledge of vector spaces and related concepts
Comment. such as linear combinations, bases, dimension, spanning, and linear dependence and independence.
Comment. For an extra challenge, try covering up the answers before attempting the question.
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Shuffle Answers.
Question. Let V be an vector space, and let W be a subset of V. What does it mean when we say that W is closed under addition?
Answer. Whenever x and y are in V, then x+y is in V.
Answer. Whenever x and y are in W, then x+y is in V.
Correct Answer. Whenever x and y are in W, then x+y is in W.
Answer. Whenever x and y are in V, then x+y is in W.
Answer. If x+y is in W, then x and y are in W.
Answer. W(x+y) = Wx + Wy for every two vectors x and y.
Answer. Every vector in W is the sum of two vectors in W.
Question. Let V be a vector space, and let W be a subset of V. What does it mean when we say that W is closed under scalar multiplication?
Answer. Whenever x is in V and c is a scalar, then cx is in V.
Answer. Whenever x is in V and c is a scalar, then cx is in W.
Correct Answer. Whenever x is in W and c is a scalar, then cx is in W.
Answer. Whenever x is in W and c is a scalar, then cx is in V.
// Thanks to Courtney Amor for correcting this answer.
Answer. If cx is in W and x is in W, then c is a scalar.
Partially Correct Answer. If cx is in W and c is a scalar, then x is in W.
Answer. W(cx) = cWx for every vector x and scalar c.
Question. Let V be a vector space, and let S be a subset of V. What does it mean when we say that S is linearly independent?
Answer. S is closed under both addition and scalar multiplication.
Answer. S is a basis.
Comment. Every basis is linearly independent, but not vice versa.
Answer. Every element in V is a linear combination of elements in S.
Partially Correct Answer. The number of elements of S is less than or equal to the dimension of V.
Comment. It is true that if S is linearly independent, then the number of elements in S is less than the dimension of V, but it is possible for S to have fewer elements than the dimension of V without S being linearly independent.
Correct Answer. The only way to write 0 as a linear combination of elements of S is the zero combination (where one takes zero multiples of each element of S).
Answer. All the elements of S are distinct from each other.
Answer. S has nullity zero.
Question. Let V be a vector space, and let S be a subset of V. What does it mean when we say that S is linearly dependent?
Answer. S is closed under both addition and scalar multiplication.
Answer. Every element of S is a linear combination of other elements of S.
Comment. Only one of the elements of S needs to be able to be expressed as a combination of the others in order to establish linear dependence.
Partially Correct Answer. The number of elements of S is greater than the dimension of V.
Correct Answer. There is a way to write 0 as a linear combination of elements of S other than the zero combination.
Answer. The span of S has smaller dimension than the dimension of V.
Answer. S depends on a linear transformation.
Answer. At least two of the elements of S are the same.
Question. Let V be a vector space, and let S be a subset of V. What does it mean when we say that S spans V?
Answer. S is a basis for V.
Answer. The elements of S are all distinct from each other.
Correct Answer. Every vector in V can be expressed as a linear combination of vectors in S.
Answer. Every vector in V has exactly one representation as a linear combination of vectors in S.
Partially Correct Answer. S has at least as many elements as the dimension of V.
Comment. It is necessary for S to have at least as many elements as the dimension of V in order for S to span V, but it is not sufficient.
Partially Correct Answer. The rank of S is the same as the dimension of V.
Comment. This is true for finite-dimensional vector spaces, but not for infinite-dimensional ones.
Question. Let V be a five-dimensional vector space, and let S be a subset of V which spans V. Then S
Correct Answer. Must consist of at least five elements.
Answer. Must have exactly five elements.
Answer. Must have at most five elements.
Answer. Must have infinitely many elements.
Answer. Must be linearly independent.
Answer. Must be a basis for V.
Answer. Must be linearly dependent.
Question. Let V be a five-dimensional vector space, and let S be a subset of V which is linearly independent. Then S
Answer. Must consist of at least five elements.
Answer. Must have exactly five elements.
Correct Answer. Must have at most five elements.
Answer. Must have infinitely many elements.
Answer. Must span V.
Answer. Must be a basis for V.
Answer. Can have any number of elements (except zero).
Question. Let V be a five-dimensional vector space, and let S be a subset of V which is linearly dependent. Then S
Answer. Must consist of at least five elements.
Answer. Must have exactly five elements.
Answer. Must have at most five elements.
Answer. Must have infinitely many elements.
Answer. Must span V.
Answer. Must be a basis for V.
Correct Answer. Can have any number of elements (except zero).
Question. Let V be a five-dimensional vector space, and let S be a subset of V which is a basis for V. Then S
Answer. Must consist of at least five elements.
Correct Answer. Must have exactly five elements.
Answer. Must have at most five elements.
Answer. Must be linearly dependent.
Correct Answer. Must span V.
Correct Answer. Must be linearly independent.
Answer. Can have any number of elements (except zero).
Question. Let V be a five-dimensional vector space, and let S be a subset of V consisting of three vectors. Then S
Correct Answer. Cannot span V, but can be linearly independent or dependent.
Answer. Must be linearly independent, but may or may not span V.
Answer. Must be linearly dependent, and must span V.
Answer. May or may not be linearly independent, and may or may not span V.
Answer. Must be linearly dependent, but may or may not span V.
Answer. Must be linearly independent, but cannot span V.
Answer. Can span V, but only if it is linearly independent, and vice versa.
Question. Let V be a three-dimensional vector space, and let S be a subset of V consisting of five vectors. Then S
Answer. Cannot span V, but can be linearly independent or dependent.
Answer. Must be linearly dependent, and must span V.
Answer. Must be linearly independent, but may or may not span V.
Answer. May or may not be linearly independent, and may or may not span V.
Correct Answer. Must be linearly dependent, but may or may not span V.
Answer. Must be linearly independent, but cannot span V.
Answer. Can span V, but only if it is linearly independent, and vice versa.
Question. Let V be a five-dimensional vector space, and let S be a subset of V consisting of five vectors. Then S
Answer. Cannot span V, but can be linearly independent or dependent.
Answer. Must be linearly dependent, and must span V.
Answer. Must be linearly independent, but may or may not span V.
Answer. Must be a basis of V.
Answer. Must be linearly dependent, but may or may not span V.
Answer. Must be linearly independent, but cannot span V.
Correct Answer. Can span V, but only if it is linearly independent, and vice versa.
Question. If v_1, v_2, v_3, v_4, v_5 are five vectors in R^3, then the number of redundant vectors
Correct Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Answer. Can be any number from zero to three.
Answer. Is three.
Question. If v_1, v_2, v_3 are three vectors in R^5, then the number of redundant vectors
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Answer. Is three.
Question. If v_1, v_2, v_3, v_4, v_5 are five vectors in R^3, then the number of non-redundant vectors
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Answer. Is three.
Question. If v_1, v_2, v_3 are three vectors in R^5, then the number of non-redundant vectors
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Answer. Is three.
Question. The rank of a 3x5 matrix
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Answer. Is three.
Question. The nullity of a 3x5 matrix
Correct Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Answer. Can be any number from zero to three.
Answer. Is three.
Question. The rank of a 5x3 matrix
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Answer. Is three.
Question. The nullity of a 5x3 matrix
Answer. Can be any number from two to five.
Answer. Must be two.
Answer. Can be any number from zero to two.
Answer. Can be any number from zero to five.
Answer. Must be zero.
Correct Answer. Can be any number from zero to three.
Comment. Thanks to Alan Jern for correcting this answer.
Answer. Is three.