Vector Spaces Quiz

Which of the following is NOT a vector space property? A) Closure under addition B) Closure under scalar multiplication C) Associativity D) Commutativity

• C) Distributivity

• A) Closure under subtraction

• B) Closure under division

• D) Commutativity

In a vector space, what is the additive identity element called? A) Zero vector B) Unit vector C) Identity vector D) Null vector

• B) Zero element

• D) Null vector

• A) Unity vector

• C) Neutral vector

What is the dimension of a vector space with a basis containing 5 vectors? A) 5 B) 10 C) 3 D) Cannot be determined

• A) 5

• B) 7

• D) 5

• C) 2

Which of the following is a subspace of a vector space? A) Plane passing through origin B) Sphere centered at origin C) Circle not passing through origin D) Line not passing through origin

• A) Plane passing through origin

• B) Sphere centered at origin

• C) Circle not passing through origin

• D) Line not passing through origin

If two vectors in a vector space are linearly dependent, what can be said about their scalar multiples? A) They are linearly independent B) They are equal C) They are parallel D) They are orthogonal

• C) They are parallel

• A) They are linearly independent

• B) They are equal

• D) They are orthogonal

What is the span of a set of vectors in a vector space? A) The set of all possible linear combinations of the vectors B) The set of all possible scalar multiples of the vectors C) The set of all possible dot products of the vectors D) The set of all possible cross products of the vectors

• A) The set of all possible linear combinations of the vectors

• B) The set of all possible scalar multiples of the vectors

• C) The set of all possible dot products of the vectors

• D) The set of all possible cross products of the vectors

Which of the following is NOT a vector space property? A) Distributivity of scalar multiplication over vector addition B) Existence of additive inverse C) Existence of multiplicative inverse D) Associativity of scalar multiplication

• D) Associativity of scalar multiplication

• B) Existence of additive inverse

• C) Existence of multiplicative inverse

• A) Distributivity of scalar multiplication over vector addition

What is the rank of a matrix representing a linear transformation from a 3-dimensional vector space to a 2-dimensional vector space? A) 3 B) 2 C) 1 D) 0

• D) 0

• C) 1

• A) 3

• B) 2

Which of the following is a basis for the vector space R^3? A) {(1,0,0), (0,1,0)} B) {(1,0,0), (0,0,1)} C) {(1,1,1), (0,1,0), (0,0,1)} D) {(1,0,0), (0,1,0), (0,0,1)}

• D) {(1,0,0), (0,1,0), (0,0,1)}

• C) {(1,0,1), (0,1,0), (0,0,1)}

• B) {(1,1,0), (0,0,1)}

• A) {(1,0,0), (0,1,1)}

What is the null space of a matrix representing a linear transformation? A) The set of all vectors that map to the zero vector B) The set of all vectors that map to a non-zero vector C) The set of all vectors that are orthogonal to the zero vector D) The set of all vectors that are parallel to the zero vector

• A) The set of all vectors that map to the zero vector

• D) The set of all vectors that are parallel to the zero vector

• B) The set of all vectors that map to a non-zero vector

• C) The set of all vectors that are orthogonal to the zero vector