2.which of the following are subspace of P_4(the set of all polynomials of degree less than 4)
(c)the set of all polynomials p(x) in P_4 such that p(0) = 0
(d)the set of all polynomials p(x) in P_4 such that p(1) = 0
(e)the set of all polynomials in P4 have least one real root
想請問一下各位，這題的答案是c.d嗎?
我是覺得e也有因為至少有個0存在
還有為什麼d選項是對的，希望有人可以幫我解答 謝謝各位
3 則留言:
(e)的反例
f(x)=x^2+2x+1
g(x)=2x
f g 皆至少有一實根
但是 f(x)-g(x)=x^2+1 只有複數根
所以不具封閉性
(d)令
W={p(x) for all p(x)屬於p4|p(1)=0}
for all f(x)屬於W =>f(1)=0
for all g(x)屬於W =>g(1)=0
因為af(x)+bg(x)屬於p4 且 當x=1時代入
=> af(1)+bg(1)=a*0+b*0=0
=> af(x)+bg(x)屬於W
=> W為p4之子空間
謝謝您幫忙解答，非常感激!
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