Math 2.0

definition

Linear span

Definition

Let FF be a field, let VV be a vector space over FF, and let S⊆VS \subseteq V. The linear span of SS is the set span⁡(S)\operatorname{span}(S) of all linear combinations of finitely many vectors of SS: span⁡(S)={a1v1+⋯+akvk:k≥0, ai∈F, vi∈S}.\operatorname{span}(S) = \{a_1 v_1 + \cdots + a_k v_k : k \ge 0, \ a_i \in F, \ v_i \in S\}.

Remarks

The span is a linear subspace of VV, and it is the intersection of all subspaces of VV that contain SS. The span of the empty set is {0}\{0\}. In R3\R^3, the span of one nonzero vector is a line through the origin. The span of two non-parallel vectors is a plane through the origin.

Explanations

L1 human

The span of some vectors is everything we can reach from them by scaling and adding. Two arrows in space that do not point along the same line span a whole plane. The span is always a subspace.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear span, definition, no evidence levelLinear spandefinition
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  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear span definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Linear_span, Definition

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