Vector Spaces, Dual Spaces and Inner Product Spaces - Question Bank

1. Which norm on R^n is derived from the standard Euclidean inner product?
A) The Euclidean norm: ||x||_2 = sqrt(x1^2 + ... + xn^2)
B) The Manhattan norm: ||x||_1 = |x1| + ... + |xn|
C) The maximum norm: ||x||_inf = max(|x1|, ..., |xn|)
D) The p-norm: ||x||_p = (|x1|^p + ... + |xn|^p)^(1/p)
2. Every inner product space is a normed vector space (with ||v|| = sqrt(<v, v>)). Is the converse true? That is, is every normed vector space an inner product space?
A) No, not all norms arise from an inner product (e.g., the max norm on R^2).
B) Yes, every norm defines a unique inner product.
C) Yes, if the norm satisfies the triangle inequality.
D) No, only if the vector space is finite-dimensional.
3. Which axiom must a norm ||.|| satisfy?
A) Positive definiteness: ||v|| >= 0, and ||v|| = 0 iff v = 0.
B) Triangle inequality: ||u+v|| <= ||u|| + ||v||.
C) Absolute homogeneity: ||cv|| = |c| ||v||.
D) All of the above.
4. What is the definition of a normed vector space?
A) A vector space V with a function ||.|| : V -> R (the norm) satisfying specific axioms.
B) A vector space V with an inner product.
C) A vector space V where every vector has a 'size'.
D) A vector space V where vectors can be added.
5. Consider the vector space of continuous functions C[a, b] on the interval [a, b]. A common inner product is defined as:
A) <f, g> = integral from a to b of f(x)g(x) dx
B) <f, g> = f(a)g(a) + f(b)g(b)
C) <f, g> = f( (a+b)/2 ) * g( (a+b)/2 )
D) <f, g> = max_{x in [a,b]} (f(x)g(x))
6. In the context of linear algebra, a 'field' is a set with two operations (addition and multiplication) that satisfy certain axioms, analogous to the properties of real or complex numbers. Which of the following is NOT typically a field used in vector spaces?
A) The set of integers (Z) under usual addition and multiplication.
B) The set of real numbers (R) under usual addition and multiplication.
C) The set of complex numbers (C) under usual addition and multiplication.
D) Finite fields like Z_p (integers modulo a prime p).
7. What is the Hilbert space theorem related to the representation of linear functionals?
A) Riesz Representation Theorem
B) Spectral Theorem
C) Hahn-Banach Theorem
D) Stone-Weierstrass Theorem
8. Which theorem states that for a self-adjoint operator on a finite-dimensional complex inner product space, eigenvectors corresponding to distinct eigenvalues are orthogonal?
A) Spectral Theorem
B) Cauchy-Schwarz Theorem
C) Riesz Representation Theorem
D) Gram-Schmidt Theorem
9. A linear operator T on an inner product space is called self-adjoint (or Hermitian if complex) if:
A) T* = T
B) T* = -T
C) T* = T^-1
D) T* = I (identity operator)
10. If T is a linear operator on a finite-dimensional inner product space, then the adjoint T* satisfies:
A) (T*)* = T
B) (T*)* = -T
C) (T*)* = T^-1
D) (T*)* = T^T
11. What is the adjoint of a linear operator T: V -> W between inner product spaces V and W?
A) An operator T*: W -> V such that <T(v), w> = <v, T*(w)> for all v in V, w in W.
B) An operator T*: V -> W such that <T(v), w> = <v, T(w)> for all v, w in V.
C) An operator T*: W -> V such that <T(v), w> = <T*(v), w>.
D) An operator T*: W -> V such that <v, T(w)> = <T*(w), v>.
12. In a finite-dimensional inner product space, if {u1, ..., un} is an orthonormal basis, then any vector v can be written as:
A) v = <v, u1>u1 + ... + <v, un>un
B) v = u1 + ... + un
C) v = <v, u1> + ... + <v, un>
D) v = (1/n) * (u1 + ... + un)
13. The process of converting an arbitrary basis into an orthonormal basis for an inner product space is called:
A) Gram-Schmidt process
B) Cauchy-Schwarz process
C) Jordan decomposition
D) Fourier series expansion
14. What is an orthonormal set of vectors?
A) An orthogonal set where each vector has a norm of 1.
B) An orthogonal set where at least one vector has a norm of 1.
C) A set of linearly independent vectors.
D) A set of orthogonal vectors.
15. An orthogonal set of non-zero vectors is always:
A) Linearly independent
B) Linearly dependent
C) A basis
D) An orthonormal set
16. Two vectors u and v in an inner product space are called orthogonal if:
A) <u, v> = 0
B) u = v
C) u = -v
D) <u, v> = 1
17. What does the Cauchy-Schwarz inequality state for an inner product space?
A) |<u, v>| <= ||u|| ||v||
B) |<u, v>| >= ||u|| ||v||
C) |<u, v>| = ||u|| ||v||
D) |<u, v>| <= ||u|| + ||v||
18. The norm (or length) of a vector v in an inner product space is defined as:
A) ||v|| = sqrt(<v, v>)
B) ||v|| = <v, v>
C) ||v|| = sqrt(<v, v> / n)
D) ||v|| = <v, 0>
19. What is the standard inner product (dot product) on R^n?
A) <x, y> = x1*y1 + x2*y2 + ... + xn*yn
B) <x, y> = x1*y1 * x2*y2 * ... * xn*yn
C) <x, y> = x1+y1 + x2+y2 + ... + xn+yn
D) <x, y> = max(xi*yi)
20. An inner product space is a vector space equipped with an:
A) Inner product
B) Outer product
C) Dot product
D) Cross product
21. For a complex vector space V, the inner product <.,.> : V x V -> C must satisfy:
A) Conjugate symmetry: <u, v> = conj(<v, u>), and linearity in the first argument.
B) Symmetry: <u, v> = <v, u>, and linearity in the first argument.
C) Linearity in both arguments.
D) Only positive-definiteness.
22. Which property is NOT required for a function <.,.> to be an inner product on a real vector space V?
A) Conjugate symmetry: <u, v> = conj(<v, u>).
B) Linearity in the first argument: <au + bv, w> = a<u, w> + b<v, w>.
C) Positive-definiteness: <v, v> >= 0, and <v, v> = 0 if and only if v = 0.
D) Symmetry: <u, v> = <v, u>.
23. What is an inner product on a real vector space V?
A) A function <.,.> : V x V -> R satisfying linearity in the first argument, symmetry, and positive-definiteness.
B) A function <.,.> : V x V -> R satisfying only positive-definiteness.
C) A function <.,.> : V x V -> R satisfying only symmetry.
D) A function <.,.> : V x V -> R that is bilinear.
24. For a finite-dimensional vector space V, there is a natural isomorphism between V and its bidual space V** (the dual of the dual space). What is this map?
A) The evaluation map: E(v)(f) = f(v)
B) The transpose map: T*(f) = f o T
C) The identity map: I(v) = v
D) The zero map: Z(v) = 0
25. What is the relationship between a basis {v1, ..., vn} of V and the dual basis {f1, ..., fn} of V*?
A) fi(vj) = δij (Kronecker delta), where δij = 1 if i=j and 0 if i≠j.
B) fi(vj) = 1 for all i, j.
C) fi(vj) = 0 for all i, j.
D) fi(vj) = i + j.
26. Let V = R^2. Which of the following is a linear functional on V?
A) f(x, y) = 3x - 2y
B) f(x, y) = x^2 + y^2
C) f(x, y) = |x| + |y|
D) f(x, y) = sin(x) + cos(y)
27. If V is a finite-dimensional vector space with dimension n, what is the dimension of its dual space V*?
A) n
B) n^2
C) 1
D) 2n
28. A linear functional is a linear map from a vector space V to its:
A) Underlying field F
B) Dual space V*
C) Itself (V)
D) R^n
29. What is a dual space V* of a vector space V?
A) The vector space of all linear functionals from V to its underlying field F.
B) The vector space of all linear transformations from V to V.
C) The vector space of all linear transformations from F to V.
D) The vector space of all non-linear functions from V to F.
30. If T: V -> W is a linear transformation, when is T invertible?
A) When T is both injective (one-to-one) and surjective (onto).
B) When T is injective.
C) When T is surjective.
D) When T maps the zero vector to the zero vector.
31. The Rank-Nullity Theorem states that for a linear transformation T: V -> W, where V is finite-dimensional:
A) dim(V) = rank(T) + nullity(T)
B) dim(W) = rank(T) + nullity(T)
C) rank(T) = nullity(T)
D) dim(V) = rank(T) - nullity(T)
32. What is the image (or range) of a linear transformation T: V -> W?
A) {w ∈ W | w = T(v) for some v ∈ V}
B) {v ∈ V | T(v) = 0_W}
C) {v ∈ V | T(v) = v}
D) {w ∈ W | T(w) = 0_W}
33. What is the kernel (or null space) of a linear transformation T: V -> W?
A) {v ∈ V | T(v) = 0_W}
B) {w ∈ W | w = T(v) for some v ∈ V}
C) {v ∈ V | T(v) != 0_W}
D) {v ∈ V | T(v) = v}
34. The set of all linear transformations from V to W forms a:
A) Vector space
B) Field
C) Group
D) Ring
35. Let V and W be vector spaces over the same field F. A function T: V -> W is a linear transformation if:
A) T(u + v) = T(u) + T(v) and T(cv) = cT(v) for all u, v ∈ V and c ∈ F.
B) T(u + v) = T(u) + T(v) for all u, v ∈ V.
C) T(cv) = cT(v) for all v ∈ V and c ∈ F.
D) T is continuous and T(0) = 0.
36. What is the dimension of the vector space of polynomials of degree at most n, Pn(x)?
A) n+1
B) n
C) 1
D) 2n
37. Consider the vector space of 2x2 matrices with real entries. What is a basis for this space?
A) {[[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[1,0]], [[0,0],[0,1]]}
B) {[[1,1],[0,0]], [[0,0],[1,1]]}
C) {[[1,0],[0,1]]}
D) {[[1,0],[0,0]]}
38. What is the dimension of the vector space R^n?
A) n
B) 1
C) n^2
D) 2^n
39. What is the dimension of a vector space V?
A) The number of vectors in any basis for V.
B) The number of vectors in any spanning set for V.
C) The number of vectors in any linearly independent set for V.
D) The number of elements in V.
40. A set of vectors that spans a vector space V and is linearly independent is called a:
A) Basis
B) Linearly dependent set
C) Generating set
D) Subspace
41. What does it mean for a set of vectors {v1, v2, ..., vn} in a vector space V to be linearly independent?
A) The only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0.
B) At least one vector vi cannot be written as a linear combination of the others.
C) The sum of the vectors is non-zero.
D) The vectors are orthogonal to each other.
42. Consider R^2. Which of the following is a subspace of R^2?
A) The set of all vectors (x, y) such that y = 2x.
B) The set of all vectors (x, y) such that x > 0.
C) The set of all vectors (x, y) such that x + y = 1.
D) The set of all vectors (x, y) such that xy = 0.
43. Which condition is NOT necessary for a non-empty subset W of a vector space V to be a subspace?
A) W is closed under addition.
B) W is closed under scalar multiplication.
C) W contains the zero vector of V.
D) W contains at least one non-zero vector.
44. What is a subspace of a vector space V?
A) A subset W of V that is itself a vector space under the same operations as V.
B) Any subset of V.
C) A subset W of V such that for any v ∈ W, kv ∈ V for any scalar k.
D) A subset W of V such that for any u, v ∈ W, u + v ∈ V.
45. Let P(x) be the set of all polynomials with real coefficients. Is P(x) a vector space over the field of real numbers R?
A) Yes, polynomial addition and scalar multiplication satisfy the vector space axioms.
B) No, the product of two polynomials is not always in P(x).
C) No, there is no additive identity element.
D) Yes, but only if the degree of the polynomials is limited.
46. Consider the set of all n-tuples of real numbers, R^n. What is the scalar multiplication operation?
A) Component-wise multiplication of the scalar with each element of the n-tuple.
B) Dot product of the scalar with the n-tuple.
C) Cross product of the scalar with the n-tuple.
D) Sum of the scalar and each element of the n-tuple.
47. Which of the following is NOT a required axiom for a set V to be a vector space over a field F?
A) Closure under addition (u + v ∈ V for all u, v ∈ V).
B) Existence of a zero vector (0 ∈ V such that v + 0 = v for all v ∈ V).
C) Associativity of scalar multiplication (a(bv) = (ab)v for all a, b ∈ F, v ∈ V).
D) Commutativity of scalar multiplication (av = va for all a ∈ F, v ∈ V).
48. What is the fundamental definition of a vector space over a field F?
A) A set V with two operations, addition and scalar multiplication, satisfying specific axioms.
B) A set V with a single operation, addition, that forms an abelian group.
C) A set V with scalar multiplication satisfying closure and distributivity.
D) A set V where elements can be added and multiplied by scalars from any field.
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