Chapter 2 Linear Vector Space
A linear vector space (also called a vector space) is a mathematical structure consisting of a set of vectors and two operations: vector addition and scalar multiplication. These operations satisfy a set of axioms that govern their behavior, including closure, associativity, commutativity, distributivity, and the existence of a zero vector and additive inverse. In a vector space, the vectors can be added and scaled by real or complex numbers, and the resulting vectors still belong to the same vector space. Additionally, vectors can be combined in linear combinations, where each vector is multiplied by a scalar coefficient and then added together. Examples of vector spaces include Euclidean space, which consists of all the possible vectors that can be represented by a list of real numbers, and function spaces, which consist of functions that can be added together and multiplied by scalars. In quantum mechanics, physical quantities are conveniently represented by linear operators in a vector (Hilbert) space. The possible quantum states of a system are described as vectors.
