# The Trivial Vector Space

The **trivial vector space (or null vector space)** contains only the zero element. $$ V = \{ \vec{0} \} $$

It's the smallest possible vector space.

**Is the null vector space an empty set?** No, the null vector space is not the empty set (V = {}). This is because an empty set does not satisfy the commutative property of vector spaces. Therefore, an empty set is not a vector space.

### Properties of the Null Vector Space

Any vector v∈V multiplied by zero results in the null space 0_{v}.

Similarly, any scalar α∈K multiplied by the null space 0_{v} results in the null space itself.

And so on.