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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA subset $S \\subseteq V$ is a *spanning set* of $V$, or *spans* $V$, when $\\operatorname{span}(S) = V$, where $\\operatorname{span}$ is the [[linear span|0DTY8B9HMS]].\nEquivalently, each vector of $V$ is a linear combination of finitely many vectors of $S$.\n\n## Remarks\n\nThe vectors $(1, 0), (0, 1), (1, 1)$ span $\\R^2$, and so do $(1, 0), (0, 1)$.\nA linearly independent spanning set is a [[basis|0KCQ9Y3QKA]].\n[[Every spanning set contains a basis|06MRNP4BNT]].\nA vector space with a finite spanning set is [[finite-dimensional|0ZHWKQMVX5]].\n\n## Explanations\n\n### L1 human\n\nA spanning set is a set of building blocks from which we can make every vector of the space.\nIt may contain more vectors than we need.\nRemoving the extra vectors leaves a basis.\n",
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