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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $S \\subseteq V$.\nThe *linear span* of $S$ is the set $\\operatorname{span}(S)$ of all [[linear combinations|0V8F3E48E6]] of finitely many vectors of $S$:\n$$\\operatorname{span}(S) = \\{a_1 v_1 + \\cdots + a_k v_k : k \\ge 0, \\ a_i \\in F, \\ v_i \\in S\\}.$$\n\n## Remarks\n\nThe span is a [[linear subspace|00YC1D652K]] of $V$, and it is the intersection of all subspaces of $V$ that contain $S$.\nThe span of the empty set is $\\{0\\}$.\nIn $\\R^3$, the span of one nonzero vector is a line through the origin.\nThe span of two non-parallel vectors is a plane through the origin.\n\n## Explanations\n\n### L1 human\n\nThe span of some vectors is everything we can reach from them by scaling and adding.\nTwo arrows in space that do not point along the same line span a whole plane.\nThe span is always a subspace.\n",
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