GPT Inner Prod Flashcards
Inner Product Space
A vector space equipped with an inner product, which is a mathematical operation that takes two vectors as input and produces a scalar as output. It satisfies properties such as linearity, conjugate symmetry, and positive definiteness.
Inner Product
A function that takes two vectors and returns a scalar, typically denoted as ⟨u, v⟩ or (u, v). It satisfies properties such as linearity, conjugate symmetry, and positive definiteness.
Linearity of Inner Product
⟨au + bv, w⟩ = a⟨u, w⟩ + b⟨v, w⟩, where a and b are scalars and u, v, w are vectors in the inner product space.
Conjugate Symmetry of Inner Product
⟨u, v⟩ = ⟨v, u⟩*, where * denotes complex conjugation. In real vector spaces, it simplifies to ⟨u, v⟩ = ⟨v, u⟩.
Positive Definiteness of Inner Product
⟨v, v⟩ ≥ 0 for all vectors v in the inner product space, and ⟨v, v⟩ = 0 if and only if v = 0 (the zero vector).
Norm
The norm of a vector v, denoted as ||v||, is the square root of the inner product of v with itself, i.e., ||v|| = √⟨v, v⟩. It represents the length or magnitude of the vector.
Orthogonal Vectors
Two vectors u and v are orthogonal if their inner product ⟨u, v⟩ = 0. Geometrically, they are perpendicular to each other.
Orthonormal Vectors
A set of vectors {v1, v2, …, vn} is orthonormal if each vector has unit norm (||vi|| = 1) and any two different vectors in the set are orthogonal (⟨vi, vj⟩ = 0 for i ≠ j).
Orthogonal Complement
The orthogonal complement of a subspace U in an inner product space is the set of all vectors that are orthogonal to every vector in U. It is denoted as U⊥.
Gram-Schmidt Process
A method for finding an orthonormal basis for a subspace by starting with a set of linearly independent vectors and applying a series of orthogonalization steps. It is useful for constructing orthogonal bases in inner product spaces.
