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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-7-18\">习题 7.18</a> \n>\n> 证明：在维数为 $n$ 的向量空间 $V$ 上，满足 $T _ { i j } = T _ { j i }$ 的秩为 2 的协变对称张量构成一个维数为 $n ( n + 1 ) / 2$ 的向量空间。\n>\n> (a) 一个类型为 $( 0 , r )$ 的张量 $S$ 称为 <u class=\"concept-term\" data-concept-identity=\"urn:uuid:9386ac37-5b97-4db1-aabf-051fc76c47d5\" data-concept-route=\"/concept-identities#concept-9386ac37-5b97-4db1-aabf-051fc76c47d5\" id=\"first-90934793b24752d4accc-zh\">全对称张量(totally symmetric tensor)</u>，如果 $S _ { i _ { 1 } i _ { 2 } \\ldots i _ { r } }$ 在任意交换指标下保持不变。在 $V ?$ 上由全对称张量张成或生成的子空间的维数是多少？\n>\n> (b) 求具有循环对称性的3阶协变张量构成的向量空间的维数\n>\n> $$\n> T (u, v, w) + T (v, w, u) + T (w, u, v) = 0.\n> $$",
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