Re: [線代] 線性獨立
※ 引述《jacky7987 (憶)》之銘言：
: Let V be a vector space over a infinite field F and v_1,...,v_n are linearly
: independent over F. Show that, for any v_1,...,v_n in V,
: u_1+a*v_1,...,u_n+a*v_n
: are linearly independent over F for all but finitely many a in F
: 感謝大家:)
你的題目有很多 v_i,....我假設 u_j 是任意的, v_i 是獨立的
We may consider W=span{u_1,...u_n,v_1,...,v_n}
and then choose a basis B={b_j} such that B contains all v_i's.
Let dimW=s and u_j=\sum_{i=1}^s a_{ji}b_j. A=(a_{ji}) is a s x n matrix.
Consider D=(d_{ji}), d_{ii}=a_{ii}+a for all i=1,..,n
We want to make D be a matrix of rank n. D'=(d_{ji}), j,i=1,...,n.
If det(D') is nonzero, then D has rank n.
But det(D') is nonzero for all but finitely many a.
QED.
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◆ From: 111.249.176.202
※ 編輯: yusd24 來自: 111.249.176.202 (10/20 23:53)
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10/21 06:03, , 1F
10/21 06:03, 1F
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