Re: [理工] 線代1題

[smartlwj (實變我好愛你)]

※ 引述《ILzi ( 並不好笑 )》之銘言：
: 改成原題目如下：
:                         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.
: 我想請問的是第二題的部份
: 再度麻煩大家了
: 謝謝


想法: 因為結論是要証出 W-per 為 T 不變子空間
      也就是說  給任意的 w in W-per 則 T(w) in W-per
      因此  只要能証明出 <u,T(w)> = 0 for all u in W  即可
      並注意到 W 是 T不變子空間為條件且 T是symmetric


証明: 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.


大致上應該是這樣  有錯請指正 謝謝

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