Re: [理工] [線代]-線性獨立／依賴五題證明題
※ 引述《ruby791104 (阿年：）)》之銘言：
: 1.Prove that any finite set of vectors that contains the zero vector muast be
: linearly dependent.
請朝L.d的定義下手
: 2.Let v1, v2 be two vectors in a vector space V. Show that v1 and v2 are
: linearly dependent if and only if one of the vectors is a scalar multiple of
: the other.
(=>)
av1+bv2=0 where a,b != 0 => av1=-bv2
(<=)
If v2=kv1 where k !=0 then
av1+bv2=av1+mv1=hv1=0 => v1 !=0 => h=0 => a=-m=-bk where a,b !=0
: 3.Prove that any nonempty subset of a linearly independent set of vectors
: {v1,…,vn} is also linearly independent.
L.i的子集還是L.i 用用看反證法吧
L.i的子集是L.d 挑選一組子集使得 av1+bv2+...+kvk=0 where a,b,..,k !=0
=>bv2+...+kvk=-av1 =>把-a除過去發現(a!=0)v1可由其他v組成 矛盾 Q.E.D
: 4.Let {v1,…,vn} be a spanning set for the vector space V, and let v be any
: other vector in V. Show that v, v1,..., vn are linearly dependent.
假設v1,..,vn已經是L.d + v 也是L.d (廢話)
假設v1,..,vn是L.i + v 變L.d
證明同3.
: 5.Let v1, v2,..., vn be linearly independent vectors in a vector space V.
: Show that v2,..., vn cannot span V.
老闆～ 反證法再來一次
大約就是 v1,..,vn span v1 <- V
但 v2,...,vn cannot span v1 Q.E.D
: 以上，麻煩好心的大大們！（鞠躬
: PS：Merry Christmas！
就將........
沒人陪的耶誕夜 竟然是線代陪我 Q___Q
我也要是L.d了啦...
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我絕對不會說 這是我的無名.........
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◆ From: 59.104.36.29
※ 編輯: chris750630 來自: 59.104.36.29 (12/24 20:33)
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12/24 23:39, , 1F
12/24 23:39, 1F
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