推 keroro321 :|R^n: 就看看是不是self-adjoint 03/31 08:44
在寫功課的時候遇到瓶頸...
找了好久找不到linear transformation是symmetric的定義...
雖然知道linear transformation和matrix是isomorphism的關係(有錯麻煩指正^^")
但實在不知道怎麼回答功課的問題
題目是
V: n-dimension subspace with the inner product <.,.>
M is a proper linear subspace of V
Define: T is a reflection across M,
i.e. if for all z in V,
z=x+y, where x is in M, y is in the orthogonal complementry of M
(y在和M垂直的那個空間)
then T(z)=x-y
Quesiton: is T symmetric? Why or why not?
然後因為對自己的數學實在很沒有信心
這題還有另外兩小題不知可否麻煩大家順便過目呢
(如有違反版規麻煩告知)
1. T is a linear transformation? Why or why not?
Sol: let z1= x1+y1, x1 is in M, y1 is in the orthogonal complementry of M
let z2= x2+y2, x2 is in M, y2 is in the orthogonal complementry of M
c: a constant
T(z1+a*z2)=T[ (x1+y1) + a*(x2+y2) ]
=T[ (x1+a*x2) + (y1+a*y2) ]
= (x1+a*x2) - (y1+a*y2)
= (x1-y1) + a*(x2-y2) = T(z1) + a*T(z2)
therefore, T is a linear transformation
2. T is a projection? Why or why not?
Sol: if T is a projection, then T 作用兩次還是T
T[ T(z) ] = T(x-y)=T(x)-T(y)=x-(-y)=x+y
but T(z)=x-y
thereofore, T is not a projection
謝謝大家幫忙!!
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 108.3.154.49