推 CMos0114:正解~~好啊 03/12 01:42
※ 引述《fatfatyu (泡麵王)》之銘言：
: 題目連結:http://uploadpie.com/2s7Kc
: 請問3(b)該如何證明才對?
t t
先說明 U={A: A = A} 和 W={A: A = -A}
是子空間(這部份你自己做)
接下來
For all A in U 交 W => A in U and A in W
t t
=> A = A and A = -A
=> A = -A
=> A = 0
所以 U 交 W = {0} t t
A + A A - A
And for all A in V , A = -------- + --------
2 2
A^t + A A + A^t A^t + A
since (---------)^t = --------- = ---------
2 2 2
A^t - A A - A^t -(A^t - A)
and (---------)^t = --------- = ----------
2 2 2
hence V = U + W
Therefore, V is direct sum of U and W
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