Vector space

A vector space over a field $\mathbb{K}$ is a set $E$ equipped with two operations :

• Vector addition $+:E\times E \rightarrow E$
• Scalar multiplication $\cdot:\mathbb{K}\times E \rightarrow E$

such that the following properties hold for all $u,v,w\in E$ and all $\lambda,\mu\in K$:

1. $E$ is an abelian group with $+$.
2. Distributivity over vector addition: $\lambda\cdot(u+v)=\lambda \cdot u+\lambda \cdot v$
3. Distributivity over scalar addition: $(\lambda+\mu)\cdot u=\lambda\cdot u+\mu\cdot u$
4. Compatibility with field multiplication: $\lambda\cdot(\mu\cdot u)=(\lambda \mu)u$
5. Identity scalar: $1\cdot u=u$ where $1$ is the multiplicative identity of $\mathbb{K}$.