Definition:Generated Submodule/Linear Span
Definition
Let $K$ be a division ring or a field.
Let $V$ be a vector space over $K$.
Let $A \subseteq V$ be a subset of $V$.
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 $A$.
The linear span of $A$ is formally defined as:
- $\map \span A = \ds \set {\sum_{i \mathop = 1}^n \alpha_i v_i: n \in \N_{\ge 1}, \alpha_i \in K, v_i \in A}$
Linear Span in Real Vector Space
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:
- $\ds \map \span {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \alpha_i \mathbf v_i: 1 \le i \le k, \alpha_i \in \R, \mathbf v_i \in A }$
Also denoted as
One also frequently encounters the notation $\sequence A$.
Typically, when $A$ is small, this is also written by substituting the braces for set notation by angle brackets.
For example, when $A = \set {x_1, x_2}$, one writes $\sequence {x_1, x_2}$ for $\span A$.
On this site, the notations using $\span$ are preferred, so as to avoid possible confusion.
Also see
Sources
- 1994: Robert Messer: Linear Algebra: Gateway to Mathematics: $\S 4.4$
- For a video presentation of the contents of this page, visit the Khan Academy.