Definition:Generated Submodule/Linear Span/Real Vector Space

Definition
Let $n \in \N_{>0}$.

Let $\R^n$ be a real vector space.

Let $A \subseteq \R^n$ be a subset of $\R^n$.

Then the linear span of $A$, denoted $\span A$ or $\map \span A$, is the set of all linear combinations (of finite length) of vectors in $A$.

In the case where $A$ is a finite subset of $\R_n$ such that:


 * $A = \set {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k}$

for some $k \in \N_{>0}$, the linear span of $A$ is formally defined as:


 * $\ds \map \span {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \alpha_i \mathbf v_i: 1 \le i \le k, \alpha_i \in \R, \mathbf v_i \in A }$