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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-5-11\">习题 5.11</a> \n>\n> 设 $V$ 为一个复向量空间，其上带有一个“<u id=\"concept-5-indefinite-inner-product\" class=\"concept-term\" data-concept-identity=\"urn:uuid:cd664333-1e91-40f6-879d-2b48d97fc6d5\" data-concept-route=\"/concept-identities#concept-cd664333-1e91-40f6-879d-2b48d97fc6d5\">不定内积(indefinite inner product)</u>”，其定义为满足 (IP1)、(IP2)，但将 (IP3) 替换为非奇异性条件的内积\n>\n> (IP3') 对所有 $v \\in V$ 均有 $\\langle u \\vert v \\rangle = 0$，蕴含 $u = 0$\n>\n> (a) 证明对于这样的不定内积，可以证明与<a href=\"01-real-inner-product-spaces#theorem-5-2\" class=\"cross-reference\" id=\"xref-02-complex-inner-product-spaces-19\" data-reference=\"theorem-5-2\">定理 5.2</a> 和 5.3 类似的结果。\n>\n> (b) 若对角线上有 $p$ 个元素等于 $+1$，且有 $q$ 个元素等于 $-1$，求标准正交基之间的变换群 $U ( p , q )$ 的定义关系。",
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