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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ and $W$ be vector spaces over $F$.\nA *linear isomorphism* from $V$ to $W$ is a bijective [[linear map|0NRYHJPQG2]] $T \\colon V \\to W$.\nThe spaces $V$ and $W$ are *isomorphic*, written $V \\cong W$, when a linear isomorphism $V \\to W$ exists.\n\n## Remarks\n\nThe inverse map $T^{-1} \\colon W \\to V$ of a linear isomorphism is linear.\nA linear map is an isomorphism if and only if it sends some basis, or each basis, of $V$ to a basis of $W$.\nFor an $n \\times n$ matrix $A$ over $F$, the map $x \\mapsto Ax$ is an isomorphism of $F^n$ if and only if $A$ is [[invertible|0EB9N0S33C]].\n[[Finite-dimensional spaces are isomorphic if and only if they have equal dimension|0VJBSK15SN]].\n\n## Explanations\n\n### L1 human\n\nA linear isomorphism is a perfect translation between two vector spaces.\nIt matches the vectors one to one and respects addition and scaling.\nIsomorphic spaces have the same linear algebra, even when their vectors look different.\n",
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