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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.