Conversely, every zero-dimensional vector space over a given field is isomorphic to the zero-vector space.
Representation as subspace
If any vector space is over a body , then there is a clearly defined neutral element in it with regard to the vector addition, the zero vector . The set then forms a subspace of , because it is not empty and closed with regard to vector addition and scalar multiplication, that is:
The space is so, like any one-element vector space isomorphic to the zero vector space and the zero vector space of the vector space is called. Since a sub-vector space must contain at least one element, the zero vector space is the smallest possible sub-vector space of a vector space. The following always applies to
the intersection of two complementary sub-vector spaces and a vector space
Sums and products
Regarding the direct sum and direct product of vector spaces affects the zero vector space as a neutral element, that is, for every vector space is considered
In the category of all vector spaces over a given body with the linear mappings as morphisms , the zero vector space is the zero object : From each vector space there is exactly one linear mapping into the zero vector space and from the zero vector space there is exactly one linear mapping in each vector space, namely the null function in each case , which is just the respective null morphism .
Zero ring , the zero vector space can always be understood as a ring and thus as an algebra
Null module , the generalization of null vector space as a module