Spanning Set

up:: 021a MOC Linear Algebra

Given a Vector Space $(V,+,\cdot )$ over a Field $\mathbb{K}$, a subset $S\subseteq V$ is called its spanning set if all vectors $v\in V$ can be written as a finite Linear Combination of vectors in $S$.

Namely,

$$\forall v\in V,\exists \{{\alpha }_i{\}}_{i=1}^n\subseteq {\mathbb{K}}^n,\{v_i{\}}_{i=1}^n\subseteq S\mid v=\sum _{i=1}^n{\alpha }_i{v}_i$$

Properties

Note that the span of a vector space’s subset is a subspace of its vector space.

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References

• Linear span - Wikipedia