Editing Linear span (section)

=== Equivalence of definitions ===
The set of all linear combinations of a subset {{mvar|S}} of {{mvar|V}}, a vector space over {{mvar|K}}, is the smallest linear subspace of {{mvar|V}} containing {{mvar|S}}.

:''Proof.'' We first prove that {{math|span ''S''}} is a subspace of {{mvar|V}}. Since {{mvar|S}} is a subset of {{mvar|V}}, we only need to prove the existence of a zero vector {{math|'''0'''}} in {{math|span ''S''}}, that {{math|span ''S''}} is closed under addition, and that {{math|span ''S''}} is closed under scalar multiplication. Letting <math>S = \{ \mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n \}</math>, it is trivial that the zero vector of {{mvar|V}} exists in {{math|span ''S''}}, since <math>\mathbf 0 = 0 \mathbf v_1 + 0 \mathbf v_2 + \cdots + 0 \mathbf v_n</math>. Adding together two linear combinations of {{mvar|S}} also produces a linear combination of {{mvar|S}}: <math>(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) + (\mu_1 \mathbf v_1 + \cdots + \mu_n \mathbf v_n) = (\lambda_1 + \mu_1) \mathbf v_1 + \cdots + (\lambda_n + \mu_n) \mathbf v_n</math>, where all <math>\lambda_i, \mu_i \in K</math>, and multiplying a linear combination of {{mvar|S}} by a scalar <math>c \in K</math> will produce another linear combination of {{mvar|S}}: <math>c(\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n) = c\lambda_1 \mathbf v_1 + \cdots + c\lambda_n \mathbf v_n</math>. Thus {{math|span ''S''}} is a subspace of {{mvar|V}}.

:It follows that <math>S \subseteq \operatorname{span} S</math>, since every {{math|'''v'''<sub>''i''</sub>}} is a linear combination of {{mvar|S}} (trivially). Suppose that {{mvar|W}} is a linear subspace of {{mvar|V}} containing {{mvar|S}}. Since {{mvar|W}} is closed under addition and scalar multiplication, then every linear combination <math>\lambda_1 \mathbf v_1 + \cdots + \lambda_n \mathbf v_n</math> must be contained in {{mvar|W}}. Thus, {{math|span ''S''}} is contained in every subspace of {{mvar|V}} containing {{mvar|S}}, and the intersection of all such subspaces, or the smallest such subspace, is equal to the set of all linear combinations of {{mvar|S}}.