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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-3-14\">例 3.14</a> \n>\n> 设 $\\mathbb { R } _ { 1 }$ 为 $\\mathbb { R } ^ { n }$ 中由形如 $\\{ ( x _ { 1 } , 0 , 0 , \\ldots , 0 ) | x _ { 1 } \\in \\mathbb { R } \\}$ 的向量组成的子空间，而 $S _ { 1 }$ 为子空间\n>\n> $$\n> S _ {1} = \\{(0, x _ {2}, x _ {3}, \\dots , x _ {n}) | (x _ {i} \\in \\mathbb {R}) \\}.\n> $$\n>\n> 于是 $\\mathbb { R } ^ { n } = \\mathbb { R } _ { 1 } \\oplus S _ { 1 }$ 。以类似方式继续，$S _ { 1 }$ 可以写成 $\\mathbb { R } _ { 2 } =$ $\\{ ( 0 , x _ { 2 } , 0 , \\dots , 0 ) \\}$ 与一个子空间 $S _ { 2 }$ 的直和。我们最终得到直和分解\n>\n> $$\n> \\mathbb {R} ^ {n} = \\mathbb {R} _ {1} \\oplus \\mathbb {R} _ {2} \\oplus \\dots \\oplus \\mathbb {R} _ {n} \\cong \\mathbb {R} \\oplus \\mathbb {R} \\oplus \\dots \\oplus \\mathbb {R}.\n> $$\n>\n若 $U$ 与 $W$ 是任意向量空间，则可以通过一种构造性方式定义它们的直和，有时称为它们的<u id=\"concept-3-external-direct-sum\" class=\"concept-term\" data-concept-identity=\"urn:uuid:2f193cba-653a-4154-850a-a379f0081576\" data-concept-route=\"/concept-identities#concept-2f193cba-653a-4154-850a-a379f0081576\">外直和(external direct sum)</u>，即令",
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