幫你寫成數學符號一點看有沒有看有~~
State
Let V be a vector space (with finite dimension) then dim(V)=dim(N(T))+dim(R(T))
where T is a linear transformation from V to W
Proof
suppose dim(V)=n dim(N(T))=m
note
m less or equal to n (since N(T) subspace in V)
let A={v_1 v_2 ....v_m} be basis for N(T)
one can extend to be a basis for V(by repalcement thm)
say A_0={v_1 v_2 ....v_m v_(m+1) .......v_n}
claim
{T(v_(m+1)).........T(v_n)} is a basis for R(T)
pf:
(span)
Let w belong to R(T)
i.e. there exist v belong to V such that T(v)=w
since {v_1 v_2 ....v_m} are basis for V
can write v=a_1(v_1)+a_2(v_2)........+a_n(v_n) uniquely
=>w=T(v)=T(a_1(v_1)+a_2(v_2)........+a_n(v_n))=a_1(T(v_1))+a_2(T(v_2))+........+a_n(T(v_n)) (since T is linear)
so w belong to span{T(v_(m+1)).........T(v_n)}
and converse direction is clearly
(linearly independent)
傳送訊息
發表於 09-12-8 15:58:01
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