Vector spaces - linear independence, bases, dual spaces, inner product spaces, linear transformations, rank - One Line Questions

1. 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)? — (10, 14)
2. What is the standard basis for the vector space R^2? — {(1, 0), (0, 1)}
3. What does it mean for two vectors u and v to be orthogonal in an inner product space? — <u, v> = 0
4. What is the triangle inequality for norms derived from an inner product? — ||u + v|| <= ||u|| + ||v||
5. What is the Cauchy-Schwarz inequality for an inner product space? — | <u, v> | <= ||u|| * ||v||
6. What is the norm (or length) of a vector v in an inner product space? — ||v|| = sqrt(<v, v>)
7. What is the dimension of the vector space R^3? — 3
8. If V is a finite-dimensional vector space of dimension n, what is the dimension of its dual space V*? — 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? — k
10. What is the dual basis of a basis {v1, ..., vn} for V? — A basis {f1, ..., fn} for V* such that fi(vj) = delta_ij (Kronecker delta).
11. What is the relationship between a basis and linear independence? — A basis is a linearly independent set that spans the space.
12. What is an inner product on a real vector space V? — A function that takes two vectors and returns a scalar, satisfying certain properties.
13. What is a linear functional? — A linear map from a vector space to the scalar field.
14. What is a basis for a vector space V? — A set of vectors that spans V and is linearly independent.
15. What is an inner product space? — A vector space equipped with an inner product.
16. What is a linear transformation between two vector spaces U and V? — 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.
17. Given a basis {v1, ..., vn} for V, how can a linear functional f in V* be uniquely determined? — By its values on the basis vectors f(v1), ..., f(vn).
18. Which of the following is the defining property of a vector space? — Closure under addition and scalar multiplication
19. Which property must an inner product <u, v> satisfy for all vectors u, v in V and scalar c? — All of the above
20. 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)? — det(T) != 0
21. 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? — dim(Im(T)) + dim(Ker(T)) = dim(U)
22. Let T: V -> W be a linear transformation. If dim(V) > dim(W), what must be true about the kernel of T? — dim(Ker(T)) > 0
23. What does it mean for a set of vectors {v1, v2, ..., vk} in a vector space V to be linearly independent? — The only solution to the equation c1*v1 + c2*v2 + ... + ck*vk = 0 is c1 = c2 = ... = ck = 0.
24. Which property distinguishes an inner product space from a general vector space? — Existence of an inner product operation.
25. If a vector space V has a finite basis, what type of vector space is it called? — Finite-dimensional
26. If a set of vectors contains the zero vector, what can be said about its linear independence? — It is always linearly dependent.
27. If a linear transformation T: V -> V is an isomorphism, what can be said about its kernel and image? — Kernel is {0}, Image is V
28. In a complex vector space, the inner product satisfies <u, v> = conj(<v, u>). What is this property called? — Conjugate symmetry
29. In a vector space V over a field F, what does the operation of scalar multiplication involve? — Multiplying a scalar from F by a vector in V
30. Consider the vector space of n x n matrices, M_{n x n}(F). What is its dimension? — n^2
31. Consider the set of all polynomials of degree at most n, P_n. Is P_n a vector space over the real numbers? — Yes, it satisfies all the axioms of a vector space.
32. Consider the zero linear transformation T: U -> V (T(u) = 0_V for all u). What is its rank and nullity? — Rank = 0, Nullity = dim(U)
33. If T: U -> V and S: V -> W are linear transformations, what is true about their composition S o T? — S o T is a linear transformation from U to W.
34. What is the relationship between the dimension of a vector space and the number of vectors in its basis? — The dimension is equal to the number of vectors in any basis.
35. What is the nullity of a linear transformation T? — The dimension of its kernel (null space).
36. What is the rank of a linear transformation T? — The dimension of its image (range).
37. What is the dimension of a vector space? — The number of vectors in a basis for the space.
38. Which of the following is NOT a subspace of R^3? — The set of all vectors (x, y, z) such that x = 1.
39. What is a dual space of a vector space V? — The set of all linear functionals on V (linear maps from V to the scalar field).
40. What is the span of a set of vectors S? — The set of all vectors that can be expressed as a linear combination of vectors in S.
41. What is the image (or range) of a linear transformation T: U -> V? — The set of all vectors v in V such that v = T(u) for some u in U.
42. What is the kernel (or null space) of a linear transformation T: U -> V? — The set of all vectors u in U such that T(u) = 0_V.
43. What is the definition of a linear combination of vectors v1, v2, ..., vk? — The expression c1*v1 + c2*v2 + ... + ck*vk for scalars c1, ..., ck.
44. What is the zero vector in the vector space of real n-tuples, R^n? — The vector (0, 0, ..., 0)
45. If V is an n-dimensional vector space, and you have a set of n linearly independent vectors in V, what can you conclude? — This set spans V and is a basis for V.
46. What is the standard inner product (dot product) on R^n? — u . v = u1*v1 + u2*v2 + ... + un*vn
47. What is the definition of a subspace W of a vector space V? — W is a non-empty subset of V that is closed under vector addition and scalar multiplication.
48. Consider the vector space of continuous functions C[0, 1]. Is this space finite-dimensional? — No, it is infinite-dimensional.
All One Line Questions
E-Book Vector spaces - linear independence, bases, dual spaces, inner product spaces, linear transformations, rank Online Test Vector spaces - linear independence, bases, dual spaces, inner product spaces, linear transformations, rank Question Bank Vector spaces - linear independence, bases, dual spaces, inner product spaces, linear transformations, rank