※ 引述《ILzi ( 并不好笑 )》之铭言： : 改成原题目如下： : n : Let V be a subspace of R ,Alinear transformation T :V→V is said to be : symmetric if (Tu,v)=(u,Tv) for all u,v€V. : Here (x.y) is the dot product of vectors x and y. : n : A subspace W of R is said to be invariant under T if Tw€W ,for all w€W. : (a)Show that T is symmetric if and only if the matrix representation of T : relative to some orthonormal basis is symmetric. : ┴ : (b)Show that if W is invariant under T, then the orthogonal complement W : of W is also invariant under T. : 我想请问的是第二题的部份 : 再度麻烦大家了 : 谢谢 想法: 因为结论是要证出 W-per 为 T 不变子空间 也就是说 给任意的 w in W-per 则 T(w) in W-per 因此 只要能证明出 <u,T(w)> = 0 for all u in W 即可 并注意到 W 是 T不变子空间为条件且 T是symmetric 证明: For all w in W-per, then we have < u, w > = 0 for all u in W (by def) ---(1) since W is T - invariant, so for any u in W => T(u) in W then put it in (1) hence, < T(u), w > = 0 for T(u) in W Now by assumption that T is symmetric, so we obtain 0 = < T(u), w > = < u, T(w) > i.e. T(w) is in W-per , since u is in W thus, W-per is T-invariant subspace. 大致上应该是这样 有错请指正 谢谢 --

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