The kernel ker("f") of a linear map consists of vectors v that are mapped to 0 in "W". The kernel and the image are subspaces of "V" and "W", respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field "F") is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms)