3.(b)
(1)証明Mn = U 直和 W
(直和要証明兩件事1.獨立子空間 以及 2.和生成)
1.証獨立子空間(U 交集 W = {0})
pf:for all A屬於 U 交集 W
T T
=> A = A 且 A = -A
=> A = -A
=> A = 0
所以 U 交集 W = {0}
2.証和生成(Mn = U + W)
pf:for all A 屬於 Mn
T T
=> A = [(A + A )/2] + [(A - A )/2]
T T T T
其中[(A + A )/2]屬於U (因為[(A + A )/2] = [(A + A )/2] )
T T T T
[(A - A )/2]屬於W (因為[(A - A )/2] = - [(A - A )/2])
(2)求dim(U)以及dim(W)
sol:1.U是收集n*n的symmetric matrix A
因此A會對稱於對角線，所以A可有 n + (1+2+3+...+n-1)個變數
因此dim(U)=n(n+1)/2
2.根據維度定理dim(Mn)=dim(U)+dim(W)
=> n*n = n(n+1)/2 + dim(W)
=> dim(W) = n*n - n(n+1)/2 = n(n-1)/2
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