Theorem: The set ({\mathbf{0}}) containing only the zero vector is a vector space over any field (\mathbb{F}).
Verification: With the only possible operations ( \mathbf{0} + \mathbf{0} = \mathbf{0}) and (c \cdot \mathbf{0} = \mathbf{0}) for all (c \in \mathbb{F}), all ten vector space axioms are trivially satisfied.
This is called the trivial or zero vector space. It has dimension 0 and is a subspace of every vector space.
It is the unique vector space of dimension 0 (up to isomorphism).
**Theorem:** The set \(\{\mathbf{0}\}\) containing only the zero vector is a vector space over any field \(\mathbb{F}\).
**Verification:** With the only possible operations \( \mathbf{0} + \mathbf{0} = \mathbf{0}\) and \(c \cdot \mathbf{0} = \mathbf{0}\) for all \(c \in \mathbb{F}\), all ten vector space axioms are trivially satisfied.
This is called the *trivial* or *zero* vector space. It has dimension 0 and is a subspace of every vector space.
It is the unique vector space of dimension 0 (up to isomorphism).