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  "body": "\n## Definition\n\nLet $F$ be a field.\nA *vector space* over $F$ is a set $V$ with two maps: an addition $V \\times V \\to V$, $(u, v) \\mapsto u + v$, and a scalar multiplication $F \\times V \\to V$, $(a, v) \\mapsto av$.\nThese axioms must hold for all $u, v, w \\in V$ and $a, b \\in F$:\n\n1. $u + (v + w) = (u + v) + w$.\n2. $u + v = v + u$.\n3. There is an element $0 \\in V$ with $v + 0 = v$ for all $v \\in V$.\n4. For each $v \\in V$ there is an element $-v \\in V$ with $v + (-v) = 0$.\n5. $a(bv) = (ab)v$.\n6. $1v = v$, where $1$ is the unit of $F$.\n7. $a(u + v) = au + av$.\n8. $(a + b)v = av + bv$.\n\nThe elements of $V$ are *vectors*, and the elements of $F$ are *scalars*.\n\n## Remarks\n\nAxioms 1 to 4 say that $(V, +)$ is an abelian group.\nThe axioms give $0v = 0$ and $(-1)v = -v$ for all $v \\in V$.\nA vector space is a [[module|0NZ8JA25WK]] over a field; Mathlib uses `Module F V` for both.\nExamples are the [[coordinate space|0XM8BZD76G]] $F^n$, [[spaces of matrices|0RE6X3BGPN]] and [[function spaces|0GZB2SN0PH]].\n\n## Explanations\n\n### L1 human\n\nA vector space is a collection of objects that we can add together and stretch by numbers.\nArrows in the plane are the standard picture, but polynomials and functions also form vector spaces.\nThe eight axioms list the rules that make the usual algebra work.\n",
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