A vector space is a mathematical concept that describes a set of objects, called vectors, that can be added together and multiplied by a scalar. It follows certain axioms, or rules, such as closure under addition and multiplication, and the existence of a zero vector and inverse elements.
A circle is a geometric shape that consists of all points in a plane that are a fixed distance, called the radius, from a given point, called the center. It can also be described as the locus of points equidistant from a fixed point.
Proving the vector space of circles is not axiomatic is important because it challenges our understanding of vector spaces and their properties. It also highlights the limitations of axiomatic systems and the need for further exploration and development of mathematical concepts.
The axioms of a vector space include closure under addition and multiplication, commutativity and associativity of addition, distributivity of scalar multiplication, the existence of a zero vector and inverse elements, and the existence of a scalar identity element.
The vector space of circles does not satisfy all of the axioms of a vector space. For example, circles do not have a well-defined addition or multiplication operation, as they cannot be added or multiplied in the traditional sense. Additionally, circles do not have a zero vector or inverse elements, as there is no "zero circle" or "negative circle" in a geometric sense.