# Vector Space

A **vector space** $V$ over a field $F$ is a set of objects ( vectors), where two conditions hold.

- Firstly,
**vector addition**of two vectors $\ket{a}, \ket{b} \in V$ will yield a third vector $\ket{a} + \ket{b} = \ket{c}$ , also contained in $V$ . - The second condition is that
**scalar multiplication**between some $\ket{a} \in V$ and some $n \in F$ , denoted by $n\ket{a}$ , is also contained within $V$ .