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  "body": "\n## Definition\n\nLet $K$ be a field, let $V$ be a vector space over $K$, and let $A$ be an [[affine space|09XNAPTJQ7]] over $V$.\nA nonempty subset $S$ of $A$ is an affine subspace if $S = p + W$ for a point $p$ of $A$ and a linear subspace $W$ of $V$. This means that\n$$S = p + W = \\{\\, p + w \\;:\\; w \\in W \\,\\}.$$\nThe subspace $W$ is the direction of $S$. It is unique: $W = \\{\\, q - r : q, r \\in S \\,\\}$. The dimension of $S$ is the dimension of $W$.\n\n## Remarks\n\nFor each point $p$ of $S$, $S = p + W$. Thus each point of $S$ can serve as the base point.\nIn $\\mathbb{R}^n$, the affine subspaces of dimension 1 are the [[lines|0MP9BCF6DP]], and those of dimension 2 are the [[planes|0V3XAMF3C1]]. Those of dimension $n - 1$ are the hyperplanes.\nTwo affine subspaces are parallel if they have the same direction.\nMathlib's `AffineSubspace` also allows the empty set. A subset $S$ is an affine subspace in this sense if $c(q - r) + s \\in S$ for all $c \\in K$ and $q, r, s \\in S$.\n\n## Explanations\n\n### L1 human\n\nAn affine subspace is a flat piece of space that need not pass through the origin. A line or a plane that has been moved away from the origin is an example. It is a linear subspace that has been shifted by a translation.\n",
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