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  "body": "\n## Proof\n\nWe show that an isomorphism sends a basis to a basis, and that a bijection between bases extends to an isomorphism.\n\n1. Suppose $V$ and $W$ are [[isomorphic|0B8T10CE7N]], and let $T \\colon V \\to W$ be a linear isomorphism. Let $n = \\dim V$, the [[dimension|03GSZ8RTZJ]], and let $(b_1, \\dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.\n2. The family $(T(b_1), \\dots, T(b_n))$ is [[linearly independent|0CRMGPAH83]]. Suppose $\\sum_i a_i T(b_i) = 0$ with $a_i \\in F$. Then $T\\left(\\sum_i a_i b_i\\right) = 0$ by linearity.\n3. $T$ is injective, so its kernel is $\\{0\\}$ by [[the kernel criterion|0BJBYCGMWE]]. Hence $\\sum_i a_i b_i = 0$, and all $a_i = 0$ because $(b_i)$ is linearly independent.\n4. The family $(T(b_1), \\dots, T(b_n))$ [[spans|088RYKAWK9]] $W$. Let $w \\in W$. Since $T$ is surjective, $w = T(v)$ for some $v = \\sum_i a_i b_i$, and then $w = \\sum_i a_i T(b_i)$.\n5. So $(T(b_1), \\dots, T(b_n))$ is a basis of $W$ with $n$ vectors. By the [[dimension theorem|0WK0Y8RH1M]], $\\dim W = n = \\dim V$.\n6. Conversely, suppose $\\dim V = \\dim W = n$. Let $(b_1, \\dots, b_n)$ be a basis of $V$ and $(c_1, \\dots, c_n)$ a basis of $W$.\n7. By [[the extension of values on a basis|0R8X9XA83A]], there is a linear map $S \\colon V \\to W$ with $S(b_i) = c_i$ for all $i$. There is also a linear map $R \\colon W \\to V$ with $R(c_i) = b_i$ for all $i$.\n8. The composite $R \\circ S \\colon V \\to V$ is linear, since $R(S(u + av)) = R(S(u)) + a R(S(v))$ for $u, v \\in V$ and $a \\in F$. It sends each $b_i$ to $R(c_i) = b_i$.\n9. The identity map of $V$ also sends each $b_i$ to $b_i$. By the uniqueness part of [[the same result|0R8X9XA83A]], $R \\circ S = \\mathrm{id}_V$.\n10. In the same way, $S \\circ R \\colon W \\to W$ fixes each $c_i$, so $S \\circ R = \\mathrm{id}_W$.\n11. By [[the two-sided inverse criterion|01VG2A0BHW]], $S$ is bijective. So $S$ is a linear isomorphism, and $V$ and $W$ are isomorphic.\n\n## Idea\n\nAn isomorphism carries a basis to a basis, so it preserves the number of basis vectors. Conversely, if two spaces have bases of the same size, matching the basis vectors one to one defines a linear map in each direction. The two maps undo each other.\n",
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