Vector spaces - linear independence, bases, dual spaces, inner product spaces, linear transformations, rank - Question Bank

1. What is the relationship between the dimension of a vector space and the number of vectors in its basis?
A) The dimension is always less than the number of vectors in a basis.
B) The dimension is always greater than the number of vectors in a basis.
C) The dimension is equal to the number of vectors in any basis.
D) The dimension is independent of the basis.
2. Consider the vector space of n x n matrices, M_{n x n}(F). What is its dimension?
A) n
B) n^2
C) 2n
D) 1
3. If V is an n-dimensional vector space, and you have a set of n linearly independent vectors in V, what can you conclude?
A) This set spans V and is a basis for V.
B) This set is linearly dependent.
C) This set does not span V.
D) This set might not be a basis.
4. What is the span of a set of vectors S?
A) The set of all vectors that are linearly independent of S.
B) The set of all vectors that are scalar multiples of vectors in S.
C) The set of all vectors that can be expressed as a linear combination of vectors in S.
D) The set S itself.
5. What is the definition of a linear combination of vectors v1, v2, ..., vk?
A) The sum v1 + v2 + ... + vk.
B) The product c1*v1 * c2*v2 * ... * ck*vk for scalars c1, ..., ck.
C) The expression c1*v1 + c2*v2 + ... + ck*vk for scalars c1, ..., ck.
D) The set {v1, v2, ..., vk}.
6. Let T: V -> W be a linear transformation. If dim(V) > dim(W), what must be true about the kernel of T?
A) dim(Ker(T)) = 0
B) dim(Ker(T)) > 0
C) Ker(T) = {0}
D) Ker(T) = W
7. What is the relationship between a basis and linear independence?
A) A basis must be linearly dependent.
B) A basis must be linearly independent.
C) A basis is a linearly independent set that spans the space.
D) Linear independence implies spanning.
8. Consider the vector space of continuous functions C[0, 1]. Is this space finite-dimensional?
A) Yes, its dimension is 1.
B) Yes, its dimension is related to the continuity.
C) No, it is infinite-dimensional.
D) No, because it is not a subspace of R^n.
9. If {v1, ..., vk} is a linearly independent set of vectors in V, and W is the subspace spanned by these vectors, what is the dimension of W?
A) 1
B) k
C) V
D) 0
10. Which of the following is NOT a subspace of R^3?
A) The set of all vectors (x, y, 0).
B) The set of all vectors (x, y, z) such that x + y + z = 0.
C) The set of all vectors (x, y, z) such that x = 1.
D) The line passing through the origin defined by (t, 2t, 3t).
11. What is the definition of a subspace W of a vector space V?
A) W is a non-empty subset of V that is closed under vector addition and scalar multiplication.
B) W is any subset of V.
C) W is a subset of V that contains the zero vector.
D) W is a subset of V that is linearly independent.
12. Let V be a vector space and T: V -> V be a linear transformation. If T is invertible, what is true about its determinant (if V is finite-dimensional)?
A) det(T) = 0
B) det(T) = 1
C) det(T) != 0
D) det(T) can be any real number.
13. If T: U -> V and S: V -> W are linear transformations, what is true about their composition S o T?
A) S o T is not necessarily a linear transformation.
B) S o T is a linear transformation from U to W.
C) S o T is a linear transformation from V to V.
D) S o T is a linear transformation from U to V.
14. What is the triangle inequality for norms derived from an inner product?
A) ||u * v|| <= ||u|| * ||v||
B) ||u + v|| <= ||u|| + ||v||
C) ||u + v|| >= ||u|| + ||v||
D) ||u + v|| = ||u|| + ||v||
15. What is the Cauchy-Schwarz inequality for an inner product space?
A) ||u + v|| <= ||u|| + ||v||
B) | <u, v> | <= ||u|| * ||v||
C) <u, v>^2 <= ||u||^2 * ||v||^2
D) ||u|| * ||v|| <= | <u, v> |
16. In a complex vector space, the inner product satisfies <u, v> = conj(<v, u>). What is this property called?
A) Linearity
B) Positive-definiteness
C) Conjugate symmetry
D) Scalar multiplication
17. Which property distinguishes an inner product space from a general vector space?
A) Existence of a zero vector.
B) Existence of additive inverses.
C) Existence of an inner product operation.
D) Finite dimensionality.
18. What is the dual basis of a basis {v1, ..., vn} for V?
A) A basis {w1, ..., wn} such that <vi, wj> = 0 for i != j.
B) A basis {f1, ..., fn} for V* such that fi(vj) = delta_ij (Kronecker delta).
C) A basis {v1, ..., vn} itself.
D) A basis {f1, ..., fn} for V* such that fi(vj) = 1 for all i, j.
19. If a linear transformation T: V -> V is an isomorphism, what can be said about its kernel and image?
A) Kernel is V, Image is {0}
B) Kernel is {0}, Image is V
C) Kernel is {0}, Image is {0}
D) Kernel is V, Image is V
20. Consider the zero linear transformation T: U -> V (T(u) = 0_V for all u). What is its rank and nullity?
A) Rank = dim(U), Nullity = 0
B) Rank = 0, Nullity = dim(U)
C) Rank = dim(V), Nullity = 0
D) Rank = 0, Nullity = dim(V)
21. What is the nullity of a linear transformation T?
A) The dimension of its image (range).
B) The dimension of its kernel (null space).
C) The dimension of the domain space.
D) The dimension of the codomain space.
22. What is the rank of a linear transformation T?
A) The dimension of its kernel.
B) The dimension of its image (range).
C) The dimension of the domain space.
D) The dimension of the codomain space.
23. The Rank-Nullity Theorem states that for a linear transformation T: U -> V, where U is finite-dimensional, what is the relationship between the dimension of the image and the dimension of the kernel?
A) dim(Im(T)) - dim(Ker(T)) = dim(V)
B) dim(Im(T)) + dim(Ker(T)) = dim(U)
C) dim(Im(T)) * dim(Ker(T)) = dim(U)
D) dim(Im(T)) / dim(Ker(T)) = dim(U)
24. What is the image (or range) of a linear transformation T: U -> V?
A) The set of all vectors u in U such that T(u) = 0_V.
B) The set of all vectors v in V such that v = T(u) for some u in U.
C) The set of all vectors u in U such that T(u) = u.
D) The set of all vectors v in V such that T(v) = 0_U.
25. What is the kernel (or null space) of a linear transformation T: U -> V?
A) The set of all vectors v in V such that T(u) = v for some u in U.
B) The set of all vectors u in U such that T(u) = 0_V.
C) The set of all vectors u in U such that T(u) = u.
D) The set of all vectors v in V such that T(v) = 0_U.
26. Let T: R^2 -> R^2 be a linear transformation. If T(1, 0) = (2, 3) and T(0, 1) = (4, 5), what is T(3, 2)?
A) (10, 14)
B) (14, 10)
C) (12, 15)
D) (6, 9)
27. What is a linear transformation between two vector spaces U and V?
A) Any function from U to V.
B) A function T: U -> V such that T(u1 + u2) = T(u1) + T(u2) and T(c*u) = c*T(u) for all u1, u2 in U and scalar c.
C) A function T: U -> V such that T(u1 + u2) = T(u1) * T(u2).
D) A function T: U -> V such that T(c*u) = c^2*T(u).
28. What is the norm (or length) of a vector v in an inner product space?
A) ||v|| = <v, v>
B) ||v|| = sqrt(<v, v>)
C) ||v|| = <v, v>^2
D) ||v|| = |<v, v>|
29. What does it mean for two vectors u and v to be orthogonal in an inner product space?
A) <u, v> = 1
B) <u, v> = 0
C) u = v
D) u = -v
30. What is the standard inner product (dot product) on R^n?
A) u . v = u1*v1 + u2*v2 + ... + un*vn
B) u . v = u1*v2 + u2*v1 + ... + un*vn
C) u . v = u1*v1 * u2*v2 * ... * un*vn
D) u . v = sqrt(u1^2 + u2^2 + ... + un^2)
31. What is an inner product space?
A) A vector space equipped with an inner product.
B) A vector space where all vectors are orthogonal.
C) A vector space with a finite number of vectors.
D) A vector space over the complex numbers.
32. Which property must an inner product <u, v> satisfy for all vectors u, v in V and scalar c?
A) Conjugate symmetry: <u, v> = conj(<v, u>)
B) Linearity in the first argument: <c*u, v> = c * <u, v>
C) Positive-definiteness: <v, v> >= 0 and <v, v> = 0 if and only if v = 0
D) All of the above
33. What is an inner product on a real vector space V?
A) A function that takes two vectors and returns a scalar, satisfying certain properties.
B) A function that takes one vector and returns a scalar.
C) A function that takes two scalars and returns a vector.
D) A function that takes two vectors and returns a vector.
34. Given a basis {v1, ..., vn} for V, how can a linear functional f in V* be uniquely determined?
A) By its value on one vector.
B) By its values on a linearly dependent set of vectors.
C) By its values on the basis vectors f(v1), ..., f(vn).
D) By its kernel.
35. What is a linear functional?
A) A linear map from a vector space to itself.
B) A linear map from a vector space to the scalar field.
C) A non-linear map from a vector space to the scalar field.
D) A map that preserves vector addition but not scalar multiplication.
36. If V is a finite-dimensional vector space of dimension n, what is the dimension of its dual space V*?
A) 1
B) n
C) n^2
D) 2n
37. What is a dual space of a vector space V?
A) The set of all vectors in V.
B) The set of all linear transformations from V to itself.
C) The set of all linear functionals on V (linear maps from V to the scalar field).
D) The set of all subspaces of V.
38. If a vector space V has a finite basis, what type of vector space is it called?
A) Infinite-dimensional
B) Finite-dimensional
C) Trivial
D) Non-trivial
39. What is the standard basis for the vector space R^2?
A) {(1, 0), (0, 1)}
B) {(1, 1), (0, 0)}
C) {(1, 0), (1, 1)}
D) {(0, 1), (1, 0), (0, 0)}
40. What is the dimension of the vector space R^3?
A) 1
B) 2
C) 3
D) Infinite
41. What is the dimension of a vector space?
A) The number of vectors in any spanning set for the space.
B) The number of vectors in any linearly independent set for the space.
C) The number of vectors in a basis for the space.
D) The number of vectors in the entire vector space.
42. What is a basis for a vector space V?
A) A set of vectors that spans V and is linearly dependent.
B) A set of vectors that spans V and is linearly independent.
C) A set of vectors that is linearly independent but does not span V.
D) A set of vectors that is linearly dependent and does not span V.
43. If a set of vectors contains the zero vector, what can be said about its linear independence?
A) It is always linearly independent.
B) It is always linearly dependent.
C) It is linearly independent if and only if the set contains only the zero vector.
D) It depends on the field over which the vector space is defined.
44. What does it mean for a set of vectors {v1, v2, ..., vk} in a vector space V to be linearly independent?
A) Every vector in V can be written as a linear combination of these vectors.
B) The only solution to the equation c1*v1 + c2*v2 + ... + ck*vk = 0 is c1 = c2 = ... = ck = 0.
C) At least one vector can be written as a linear combination of the others.
D) The set spans the entire vector space.
45. Consider the set of all polynomials of degree at most n, P_n. Is P_n a vector space over the real numbers?
A) No, because the sum of two polynomials of degree at most n can have a degree greater than n.
B) No, because scalar multiplication is not always defined.
C) Yes, it satisfies all the axioms of a vector space.
D) Yes, but only if n=0.
46. What is the zero vector in the vector space of real n-tuples, R^n?
A) The vector (1, 1, ..., 1)
B) The vector (0, 0, ..., 0)
C) The vector (n, n, ..., n)
D) The vector (infinity, infinity, ..., infinity)
47. In a vector space V over a field F, what does the operation of scalar multiplication involve?
A) Multiplying a vector by another vector
B) Multiplying a scalar from F by a vector in V
C) Adding two vectors from V
D) Subtracting a vector from another vector
48. Which of the following is the defining property of a vector space?
A) Closure under addition and scalar multiplication
B) Existence of a multiplicative identity
C) Existence of additive inverses
D) Associativity of scalar multiplication
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