作者fakeQQQ (Phakither)
看板Grad-ProbAsk
標題Re: [理工] 線代
時間Tue Mar 17 19:25:24 2009
※ 引述《ILzi ( 並不好笑 )》之銘言：
: nxn
: 設A€F ,試證：dim(span{I,A,A^2,...})≦n
: 麻煩各位了
: 謝謝
det (A-A)= Σ(Iλi-A) = Σai*A^i = 0
Σai*A^(i+n)=0 for any n € Ν .............................(1)
[I,A,A^2,...]x=0 has the solution of x=(a0 a1 .......an) ....(2)
(2) ∴ dim (span[I,A,A^2,...A^n]) ≦ n (=n when ai=0 for all i)
(1)and(2) ∴ any A^(n+1) is linear combinition of [A,A^2,....A^n]
=>dim (span[I,A,A^2,...]) ≦ n
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