[線代] dimension, rank+nullity problem

[ssss50201 (ssss)]

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
推 LPH66       :W 的 basis 是 {x(x+1)(x-1)}
→ ssss50201   :但是從答案推 dim(W)=1 耶~ 頗確定dim(V)=4的....
推 ChocoRs     :1f答案的dim不就等於1嗎，原PO的問題是?
→ k6416337    :原PO解釋一下你怎推得nullity(T)=3的?
→ ssss50201   :經樓上的提醒我想了一下 nullity=1,
→ ssss50201   :從P(0)=P(1)=P(-1)=0.知道kenel space有0,-1,1
→ ssss50201   :所以dim[kernel]=1, 因為rank=0, 所以dim(W)=1
→ ssss50201   :是這樣嗎@_@
→ ssss50201   :kernel={0,-1,1}=span{1},故nullity=1