What are the objects in a vector space that satisfies all defined axioms generally referred to as?

In the context of vector spaces, the objects that fulfill all defined axioms are referred to as vectors. A vector space is defined over a field, typically consisting of scalars, and the vectors can be thought of as elements that can be added together and multiplied by scalars while adhering to specific properties such as closure, associativity, and distributivity.

Vectors can represent data in various forms, including geometric representations, where vectors can describe directions and magnitudes in space, or algebraically, where they are often represented as ordered lists of numbers (components). The axioms associated with a vector space ensure that operations such as addition and scalar multiplication behave consistently and predictably, enabling the mathematical structure necessary to perform linear transformations, solve systems of equations, and more within the framework of linear algebra.

Scalars, matrices, and tuples represent different mathematical objects; while they can all be associated with vectors in various contexts (e.g., scalars can be used to scale vectors, matrices can act on vectors to transform them, and tuples can represent vectors in a coordinate system), they do not intrinsically embody the properties or definitions of a vector space as vectors do. Thus, the most accurate term for the entities within a vector space is indeed vectors.