[線代] dimension, rank+nullity problem

看板Math作者 (ssss)時間13年前 (2012/11/06 12:26), 編輯推噓2(207)
留言9則, 4人參與, 最新討論串1/1
又來麻煩大家了... Let V be the vector space, under the usual operations, of real polynomials that are of degree at most 3. Let W be the subspace of all polynomials p(x) in V such that P(0)=P(1)=P(-1)=0. Then dim V + dim W is 5. 我知道dim V =4 b/c basis for V is {1,x,x^2, x^3} T:W->R(real numbers) nullity(T)=3 b/c P(0)=P(1)=P(-1)=0 rank(T)=0 b/c T把所有W裡面的多項式送到0 那這樣dim W是4, total is 9. 為什麼呢>"< -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 108.3.154.49

11/06 12:47, , 1F
W 的 basis 是 {x(x+1)(x-1)}
11/06 12:47, 1F

11/06 13:16, , 2F
但是從答案推 dim(W)=1 耶~ 頗確定dim(V)=4的....
11/06 13:16, 2F

11/06 15:13, , 3F
1f答案的dim不就等於1嗎,原PO的問題是?
11/06 15:13, 3F

11/06 15:15, , 4F
原PO解釋一下你怎推得nullity(T)=3的?
11/06 15:15, 4F

11/06 20:38, , 5F
經樓上的提醒我想了一下 nullity=1,
11/06 20:38, 5F

11/06 20:39, , 6F
從P(0)=P(1)=P(-1)=0.知道kenel space有0,-1,1
11/06 20:39, 6F

11/06 20:40, , 7F
所以dim[kernel]=1, 因為rank=0, 所以dim(W)=1
11/06 20:40, 7F

11/06 20:40, , 8F
是這樣嗎@_@
11/06 20:40, 8F

11/06 20:42, , 9F
kernel={0,-1,1}=span{1},故nullity=1
11/06 20:42, 9F
文章代碼(AID): #1Gc95xuU (Math)