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Span

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Definition
LinearAlgebra

We define the Span of v1,...,vkv_1,...,v_k to be the set of all linear combinations of v1,...,vkv_1,...,v_k and we denote this set: span(v1,...,vk)span(v_1,...,v_k) In mathematical notation this is written: span(v1,...,vk):={a1v1+...+akvk∣ai∈F,i=1,...,k}span(v_1,...,v_k):=\{a_1v_1+...+a_kv_k|a_i\in\mathbb{F},i=1,...,k\}

When span(v1,..,vk)=Vspan(v_1,..,v_k)=V, we say “v1,..,vk span Vv_1,..,v_k \text{ span } V” or “v1,...,vkv_1,...,v_k is a spanning set for VV”.

When we say the “smallest vector space containing the vectors v1,...,vkv_1,...,v_k” we mean that if W⊂VW\subset V is another subspace and va,..,vk∈Wv_a,..,v_k\in W, then span(v1,...,vk)⊂Wspan(v_1,...,v_k)\subset W.

Linked from

Span

Every Spanning Set Contains a Basis

Linear Independent Sets are Smaller than Spanning Sets

Span of Vectors is a Subspace of the Vector Space