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        "quote": "> [!exercise] <a class=\"object-permalink\" href=\"#problem-3-6\">习题 3.6</a> \n>\n> 设 $V = U \\oplus W$，并设 $v = u + w$ 为向量 $v$ 分解为来自 $u \\in U$ 与 $w \\in W$ 的向量之和的唯一分解。定义投影映射 $P _ { U } : V \\to U$ 与 $P _ { W } : V \\to$ $W$ 为\n>\n> $$\n> P _ {U} (v) = u, \\qquad P _ {W} (v) = w.\n> $$\n>\n> 证明\n>\n> (a) $P _ { U } ^ { 2 } = P _ { U }$ 与 $P _ { W } ^ { 2 } = P _ { W }$\n>\n> (b) 证明：若 $P : V \\to V$ 是满足 $P ^ { 2 } = P$ 的算符，称为<u id=\"concept-3-idempotent-operator\" class=\"concept-term\" data-concept-identity=\"urn:uuid:57d0ee24-a7ce-4e13-a210-4cc1e3b00b56\" data-concept-route=\"/concept-identities#concept-57d0ee24-a7ce-4e13-a210-4cc1e3b00b56\">幂等算符(idempotent operator)</u>，则存在子空间 $U$，使得 $P = P _ { U } . [ H i n t ;$ 令 $U = \\{ u \\vert P u = u \\}$ 和 $\\mathcal { W } = \\{ w \\vert P w = 0 \\}$，并证明它们是互补子空间，使得 $P = P _ { U }$ 和 $P _ { \\cal W } = \\mathrm { i d } _ { V } - P . ]$",
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