Research on the Vector Space and Its Linearity

A vector space is an abelian group over a field. From the viewpoint of group theory, the tangent space is a theoretical concept. Tangent space is similar to a vector space, particularly in terms of linearity. From a field's perspective, a subfield is a subset of the original field that shares the same properties as the original field. Subfield is logically similar to a subspace of a vector space. A vector is an element of a vector space. Weight vectors can be used to describe motion, which can be applied in pattern recognition. A linear combination of weight vectors can describe a sequence of motions. Kernel graph theory treats a graph as an element of its theoretical structure. Kernel graph theory can also be applied in pattern recognition. So, basically, this article presents a perspective on applying vector space in pattern recognition. The inherent logic of vector space can be applied to both motion synthesis and graph pattern recognition. The linearity of the vector space is the main focus of this article.