Zero vector space

The zero vector space (also zero space ) in mathematics is a vector space that consists of only one vector , the zero vector . Except for isomorphism, the zero vector space is the only vector space with dimension and its basis is the empty set . Each vector space contains the zero vector space as the smallest possible sub-vector space . With regard to the direct sum and the direct product of vector spaces, the zero vector space acts as a neutral element . In the category of vector spaces over a given body , the null vector space is the null object . ${\displaystyle 0}$

definition

The zero vector space is a vector space over an arbitrary field consisting of the one-element set provided with the only possible vector addition given by ${\displaystyle (\{0\},+,\cdot )}$ ${\displaystyle K}$ ${\displaystyle \{0\}}$

${\displaystyle 0+0=0}$

and the only possible scalar multiplication given by

${\displaystyle \alpha \cdot 0=0}$

for all scalars . The vector is thus the neutral element with regard to vector addition and is called the zero vector . ${\displaystyle \alpha \in K}$ ${\displaystyle 0}$

properties

Vector space axioms

The zero vector space satisfies the axioms of a vector space :

• ${\displaystyle (\{0\},+)}$is an Abelian group , namely the trivial group
• the associative and distributive laws of scalar multiplication apply, i.e. for all : ${\displaystyle \alpha ,\beta \in K}$
  • ${\displaystyle \alpha \cdot (\beta \cdot 0)=\alpha \cdot 0=0=(\alpha \cdot \beta )\cdot 0}$
  • ${\displaystyle \alpha \cdot (0+0)=\alpha \cdot 0=0=0+0=\alpha \cdot 0+\alpha \cdot 0}$
  • ${\displaystyle (\alpha +\beta )\cdot 0=0=0+0=\alpha \cdot 0+\beta \cdot 0}$
• the unity is neutral: ${\displaystyle 1\in K}$
  • ${\displaystyle 1\cdot 0=0}$

Basis and dimension

The only basis of the zero vector space is the empty set , because we have for the linear hull of the empty set

${\displaystyle \langle \emptyset \rangle =\{0\}}$.

The dimension of the zero vector space is thus

${\displaystyle \dim(\{0\})=|\emptyset |=0}$.

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: ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle 0_{V}}$ ${\displaystyle U=\{0_{V}\}}$ ${\displaystyle V}$

• ${\displaystyle U\neq \emptyset }$
• ${\displaystyle 0_{V}+0_{V}=0_{V}\in U}$
• ${\displaystyle \alpha \cdot 0_{V}=0_{V}\in U}$ for all ${\displaystyle \alpha \in K}$

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 ${\displaystyle \{0_{V}\}}$ ${\displaystyle V}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle V}$

${\displaystyle U_{1}\cap U_{2}=\{0_{V}\}}$.

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 ${\displaystyle V}$

${\displaystyle \{0\}\oplus V\cong V\cong V\oplus \{0\}}$   or   . ${\displaystyle \{0\}\pi V\cong V\cong V\pi \{0\}}$

For the tensor product, on the other hand, it acts as an absorbing element , that is

${\displaystyle \{0\}\otimes V\cong \{0\}\cong V\otimes \{0\}}$.

Category theory

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 .

See also

• 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

literature

• Gilbert Strang : Linear Algebra . Springer, Berlin et al. 2003, ISBN 3-540-43949-8 .