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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a [[vector space|000AYPA9AJ]] over $F$.\nA *linear subspace* of $V$ is a subset $W \\subseteq V$ with these three properties:\n\n1. $0 \\in W$.\n2. If $u, v \\in W$, then $u + v \\in W$.\n3. If $a \\in F$ and $v \\in W$, then $av \\in W$.\n\n## Remarks\n\nWith the operations of $V$, a subspace $W$ is itself a vector space over $F$.\nThe sets $\\{0\\}$ and $V$ are subspaces of $V$.\nThe lines through the origin are the one-dimensional subspaces of $\\R^2$; a line that misses the origin is not a subspace.\nThe kernel and the image of a linear map are subspaces.\n\n## Explanations\n\n### L1 human\n\nA subspace is a smaller vector space inside a bigger one.\nWe cannot leave it by adding its vectors or by scaling them.\nIn space, the subspaces are the origin, the lines and planes through the origin, and the whole space.\n",
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