Vector Spaces, Dual Spaces and Inner Product Spaces - One Line Questions
1.
If T is a linear operator on a finite-dimensional inner product space, then the adjoint T* satisfies: —
(T*)* = T
2.
Consider the vector space of 2x2 matrices with real entries. What is a basis for this space? —
{[[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[1,0]], [[0,0],[0,1]]}
3.
What is the kernel (or null space) of a linear transformation T: V -> W? —
{v ∈ V | T(v) = 0_W}
4.
What is the image (or range) of a linear transformation T: V -> W? —
{w ∈ W | w = T(v) for some v ∈ V}
5.
Consider the vector space of continuous functions C[a, b] on the interval [a, b]. A common inner product is defined as: —
<f, g> = integral from a to b of f(x)g(x) dx
6.
Two vectors u and v in an inner product space are called orthogonal if: —
<u, v> = 0
7.
What is the standard inner product (dot product) on R^n? —
<x, y> = x1*y1 + x2*y2 + ... + xn*yn
8.
What does the Cauchy-Schwarz inequality state for an inner product space? —
|<u, v>| <= ||u|| ||v||
9.
The norm (or length) of a vector v in an inner product space is defined as: —
||v|| = sqrt(<v, v>)
10.
What is an inner product on a real vector space V? —
A function <.,.> : V x V -> R satisfying linearity in the first argument, symmetry, and positive-definiteness.
11.
What is the fundamental definition of a vector space over a field F? —
A set V with two operations, addition and scalar multiplication, satisfying specific axioms.
12.
What is a subspace of a vector space V? —
A subset W of V that is itself a vector space under the same operations as V.
13.
What is the definition of a normed vector space? —
A vector space V with a function ||.|| : V -> R (the norm) satisfying specific axioms.
14.
What is the adjoint of a linear operator T: V -> W between inner product spaces V and W? —
An operator T*: W -> V such that <T(v), w> = <v, T*(w)> for all v in V, w in W.
15.
What is an orthonormal set of vectors? —
An orthogonal set where each vector has a norm of 1.
16.
A set of vectors that spans a vector space V and is linearly independent is called a: —
Basis
17.
Which of the following is NOT a required axiom for a set V to be a vector space over a field F? —
Commutativity of scalar multiplication (av = va for all a ∈ F, v ∈ V).
18.
Consider the set of all n-tuples of real numbers, R^n. What is the scalar multiplication operation? —
Component-wise multiplication of the scalar with each element of the n-tuple.
19.
For a complex vector space V, the inner product <.,.> : V x V -> C must satisfy: —
Conjugate symmetry: <u, v> = conj(<v, u>), and linearity in the first argument.
20.
Which property is NOT required for a function <.,.> to be an inner product on a real vector space V? —
Conjugate symmetry: <u, v> = conj(<v, u>).
21.
The Rank-Nullity Theorem states that for a linear transformation T: V -> W, where V is finite-dimensional: —
dim(V) = rank(T) + nullity(T)
22.
Let V = R^2. Which of the following is a linear functional on V? —
f(x, y) = 3x - 2y
23.
What is the relationship between a basis {v1, ..., vn} of V and the dual basis {f1, ..., fn} of V*? —
fi(vj) = δij (Kronecker delta), where δij = 1 if i=j and 0 if i≠j.
24.
The process of converting an arbitrary basis into an orthonormal basis for an inner product space is called: —
Gram-Schmidt process
25.
An inner product space is a vector space equipped with an: —
Inner product
26.
An orthogonal set of non-zero vectors is always: —
Linearly independent
27.
What is the dimension of the vector space R^n? —
n
28.
If V is a finite-dimensional vector space with dimension n, what is the dimension of its dual space V*? —
n
29.
What is the dimension of the vector space of polynomials of degree at most n, Pn(x)? —
n+1
30.
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? —
No, not all norms arise from an inner product (e.g., the max norm on R^2).
31.
Which axiom must a norm ||.|| satisfy? —
All of the above.
32.
What is the Hilbert space theorem related to the representation of linear functionals? —
Riesz Representation Theorem
33.
Which theorem states that for a self-adjoint operator on a finite-dimensional complex inner product space, eigenvectors corresponding to distinct eigenvalues are orthogonal? —
Spectral Theorem
34.
Let V and W be vector spaces over the same field F. A function T: V -> W is a linear transformation if: —
T(u + v) = T(u) + T(v) and T(cv) = cT(v) for all u, v ∈ V and c ∈ F.
35.
A linear operator T on an inner product space is called self-adjoint (or Hermitian if complex) if: —
T* = T
36.
Which norm on R^n is derived from the standard Euclidean inner product? —
The Euclidean norm: ||x||_2 = sqrt(x1^2 + ... + xn^2)
37.
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? —
The evaluation map: E(v)(f) = f(v)
38.
What is the dimension of a vector space V? —
The number of vectors in any basis for V.
39.
What does it mean for a set of vectors {v1, v2, ..., vn} in a vector space V to be linearly independent? —
The only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0.
40.
Consider R^2. Which of the following is a subspace of R^2? —
The set of all vectors (x, y) such that y = 2x.
41.
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? —
The set of integers (Z) under usual addition and multiplication.
42.
What is a dual space V* of a vector space V? —
The vector space of all linear functionals from V to its underlying field F.
43.
A linear functional is a linear map from a vector space V to its: —
Underlying field F
44.
In a finite-dimensional inner product space, if {u1, ..., un} is an orthonormal basis, then any vector v can be written as: —
v = <v, u1>u1 + ... + <v, un>un
45.
The set of all linear transformations from V to W forms a: —
Vector space
46.
Which condition is NOT necessary for a non-empty subset W of a vector space V to be a subspace? —
W contains at least one non-zero vector.
47.
If T: V -> W is a linear transformation, when is T invertible? —
When T is both injective (one-to-one) and surjective (onto).
48.
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? —
Yes, polynomial addition and scalar multiplication satisfy the vector space axioms.