Anonymous
Anonymous asked in 科學數學 · 1 decade ago

管數問題!(span R space)

搞不太清楚span R space 和 bases for R的觀念!

例如以下例題要如何求解呢?

1.求(1,2,-1),(6,3,0),(4,-1,2),(2,-5,4)可否span R3?

麻煩各位幫忙...謝謝喔!!!

1 Answer

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  • 1 decade ago
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    1. A(1,2,-1)+B(6,3,0)+C(4,-1,2) = 0

    A+6B+4C=0

    2A+3B-C=0

    -A+2C=0

    => A=2C

    => B=-C

    2C-6C+4C=0 => 0=0

    所以C為任意數 線性相依(NG)

    2. A(1,2,-1)+B(6,3,0)+C(2,-5,4)=0

    A+6B+2C=0

    2A+3B-5C=0

    -A+4C=0

    => A=4C

    => B=-C

    4C-6C+2C=0 => 0=0

    所以C為任意數 線性相依(NG)

    3. A(1,2,-1)+B(4,-1,2)+C(2,-5,4)=0

    A+4B+2C=0

    2A-B-5C=0

    -A+2B+4C=0

    B=-C & A=2C

    4C+C-5C=0 => 0=0

    所以C為任意數 線性相依(NG)

    4. A(6,3,0)+B(4,-1,2)+C(2,-5,4)=0

    6A+4B+2C=0

    3A-B-5C=0

    2B+4C=0

    B=-2C

    A=C

    6C-4C+2C=0 => 0=0

    所以C為任意數 線性相依(NG)

    皆為線性相依, 故答案是 "否" !

    重點:

    若三向量 x , y , z 可以SPAN R3

    則該三向量必須線性獨立!

    ie, 若 Ax + By +Cz = 0, 則 A=B=C=0

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