[線代] T為W的垂直投影則I-T在W^perp也是

看板Math作者znmkhxrw (QQ)時間2年前 (2021/06/18 19:05)推噓6(6推 0噓 20→)

留言26則, 4人參與討論串1/1

想請問一下Friedberg書中的習題： -------------------------------------------------------------------- Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal projection on W, then I-T is the orthogonal projection on W^perp -------------------------------------------------------------------- 我這邊完全採用Friedberg的定義下去做, 結果證明的過程不需要dim(W)<infinity 但是以我看這本書的經驗, 他幾乎沒有過給多餘條件的情況...所以才懷疑我證錯下面我會給他的定義以及我的證明, 再請各位板友幫忙, 謝謝！ P.S. 我猜測他給dim(W)<infinity是為了垂直投影的存在性但是就他題目敘述的邏輯, 他存在性已經用假設的了, 那就根本不需要對W做假設 ========================================================================= 【定義】(R(T)是Range of T, N(T)是Null space of T) Let V be an inner product space over F = R or C Then (1) we say a linear transformation T: V→V is a projection if there exists subspaces W_1, W_2 s.t. V=W_1⊕W_2 and for any x = x_1 + x_2 with x_1€W_1 and x_2€W_2, we have T(x) = x_1 Moreover, we say T is a projection on W_1 (2) we say a projection T: V→V is an orthogonal projection if R(T)^perp = N(T) and N(T)^perp = R(T) 【性質】(用不到這個, 只是wiki跟書上都說是等價的也好證所以列一下) Let V be an inner product space over F = R or C and T: V→V be a linear transformation Then T is a projection <=> T^2 = T ！！注意到以上這些完全不用對V跟W做任何維度假設！！【想證】 Let V be an inner product space over F = R or C and W be a subspace of V (不需要finite-dimensional) if T is the orthogonal projection on W then I-T is the orthogonal projection on W^perp pf: Since T is the orthogonal projection on W, we have (1) V = W⊕W_2 and for any x with x = x_1 + x_2, x_1€W, x_2€W_2 we have T(x) = x_1 (2) R(T)^perp = N(T) and N(T)^perp = R(T) (3) From (1), (2) we have R(T) = W, N(T) = W_2 and W^perp = W_2 and W_2^perp = W (其實這裡就能證出一個性質：(W^perp)^perp = W, 不過這裡用不到) Now consider U := I-T Then (a) U(x) = (I-T)(x) = x - x_1 = x_2€W_2 (b) R(U) = {(I-T)(x)} = {x-x_1} = {x_2} = W_2 (c) N(U) = {x│U(x)=0} = {x│x_2 = 0} = W (d) R(U)^perp = W_2^perp = W = N(U) (e) N(U)^perp = W^perp = W_2 = R(U) Hence from (a)~(e), we know U is an orthogonal projection on W^perp, 證畢 -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 123.110.132.77 (臺灣) ※ 文章網址: https://www.ptt.cc/bbs/Math/M.1624014345.A.B81.html ※ 編輯: znmkhxrw (123.110.132.77 臺灣), 06/18/2021 19:06:18

→