Vector subspace

Let $V$ be a vector space.

Set $U$ is a vector subspace of $V$ generated by $U$ if:

$$U=\operatorname{span}\{\vec u_1,\dots,\vec u_n\} =\left\{ \sum_{i=1}^n \lambda_i \vec u_i \;\middle|\; \lambda_i \in \mathbb{F} \right\}$$

for $\vec u_1,\cdots, \vec u_n \in V$.

Then Set $U$ is a vector subspace of $V$.

Set $\{u_1,\dots,u_n\}$ is then known as a generating set of $U$