= Linear span
{title2=$\operatorname{span}(S)$}
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= Span
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The linear span of a subset $S$ of a <vector space> is the <set> of all finite <linear combinations> of elements of $S$. It is the smallest <vector subspace> containing $S$: every <vector subspace> containing the original <vectors> must contain their sums and <scalar multiples>.
In $\mathbb R^3$, the <linear span> of $(1,0,0)$ is a line, the <linear span> of $(1,0,0),(0,1,0)$ is the plane $z=0$, and adding $(0,0,1)$ gives all of $\mathbb R^3$. Adding $(2,0,0)$ to the first set changes none of its attainable <vectors>, because it is already a <scalar multiple> of $(1,0,0)$. Thus a <spanning set> can contain redundant generators; <linear independence> identifies when this redundancy is absent. In an infinite-dimensional space, an algebraic <linear span> still uses finite sums; a <closed linear span> can also include <limits> in a specified <norm>.
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