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        "quote": "\n\n> [!theorem] <a class=\"object-permalink\" href=\"#theorem-3-5\">定理 3.5</a> · 基下展开的唯一性 (Uniqueness of expansion in a basis)\n>\n> 设 $V$ 为有限维向量空间，$n=\\dim V$。若 $\\{ e _ { 1 } , \\ldots , e _ { n } \\}$ 是 $V$ 的基，则每个向量 $v \\in V$ 都有唯一的分解\n>\n> <span id=\"equation-3-5\" class=\"block-anchor\"></span>\n>\n> $$\n> v = \\sum_ {i = 1} ^ {n} v ^ {i} e _ {i}, \\quad v ^ {i} \\in \\mathbb {K}.\\tag{3.5}\n> $$\n>\n> $n$个标量$\\boldsymbol { v } ^ { i } \\in \\mathbb { K }$称为向量$v$在此基下的<u id=\"concept-3-component\" class=\"concept-term\" data-concept-identity=\"urn:uuid:a7f1665c-c377-4cb5-8778-59e8e0abffc1\" data-concept-route=\"/concept-identities#concept-a7f1665c-c377-4cb5-8778-59e8e0abffc1\">分量(component)</u>。\n\n> [!proof]- 证明\n>\n> 由于 $e _ { i }$ 张成 $V$，每个向量 $v$ 都具有形如<a href=\"#equation-3-5\" class=\"cross-reference\" id=\"xref-05-bases-of-a-vector-space-5\" data-reference=\"equation-3-5\">式 (3.5)</a> 的分解。若存在第二种这样的分解，\n>\n> $$\n> v = \\sum_ {i = 1} ^ {n} v ^ {i} e _ {i} = \\sum_ {i = 1} ^ {n} w ^ {i} e _ {i}\n> $$\n>\n> 那么\n>\n> $$\n> \\sum_ {i = 1} ^ {n} (v ^ {i} - w ^ {i}) e _ {i} = 0.\n> $$\n>\n> 由于 $e _ { i }$ 线性无关，该和式中的每个系数都必须为零，$v ^ { i } - w ^ { i } = 0$ 因此 $\\boldsymbol { v } ^ { i } = \\boldsymbol { w } ^ { i }$，且分解是唯一的。\n",
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