推 wheels:黃子嘉上課講義好像有這題，應該沒錯！ 11/24 21:34
推 ntust661:只能推了 11/24 21:40
推 wheels:(a)的最後x!=0 => Ax!=0 則 Ax=0 => x=0 所以A:nonsingular 11/24 21:40
→ wheels:狗尾續貂一下XD 11/24 21:41
推 askaleroux:感謝 11/25 23:19
(a)
x =/= 0 , <x,x> = (Ax)^T(Ax) > 0 , 令 Ax = [ x1,x2,x3 ... xn]^T
<x,x> = x1^2 + x2^2 + ... + xn^2 > 0
=> x1,x2...xn 不全為0
也就是Ax =/= 0
A若可逆則Ax = 0 只有當x = 0才成立
因此A要可逆
(b)
A : n*n
x : n*1
Ax : n*1
(Ax)^T : 1*n
(Ax)^T(Ay) : 1*1 ，所以(Ax)^T(Ay) = ((Ax)^T(Ay))^T
希望沒錯QQ
※ 引述《askaleroux (aska)》之銘言：
: Let the elements of an inner product space V be all vectors in R^n
: Define as
: <x,y> = (Ax)^T(Ay) where A is an n*n matrix
: (a) What is the most genera; matrix A such that above is an inner product
: => A必須可逆(Why????)
: (b) What are the properties stated in (a) ?
: 其中
: <x,y> = (Ax)^T(Ay) = ((Ax)^T(Ay))^T = (Ay)^T(Ax)
: 為什麼忽然加了一個轉置還會相等？
: =============================================================
: 有請大大姐答
: 謝謝
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