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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.\nThen $W$ is finite-dimensional, and $\\dim W \\le \\dim V$, where $\\dim$ is the [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nA basis of $W$ is a linearly independent family in $V$, so it has at most $\\dim V$ vectors.\nEquality holds only when $W = V$, see [[a subspace of full dimension is the whole space|0WGAXEK99E]].\nFor infinite-dimensional $V$, the inequality $\\dim W \\le \\dim V$ of cardinals still holds.\n\n## Explanations\n\n### L1 human\n\nA subspace cannot have more independent directions than the space around it.\nA plane in ordinary space has dimension $2$, which is at most $3$.\nThe only subspace with the same dimension is the whole space.\n",
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