基本介紹
- 中文名:數乘變換
- 外文名:transformation of scalar multiplicalion
- 所屬學科:數學
- 所屬問題:線性代數(向量空間)
- 簡介:一種線性變換
基本介紹,線性變換的概念,數乘變換的概念,相關性質,
基本介紹
線性變換的概念
設V為數域F上的線性空間,
是V到V的一個映射(變換),且滿足條件:
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(1)對任意的α,β∈V有:
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(2)對任意的α∈V及任意的實數k∈F,有:
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則稱
為V的線性變換。
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設V是數域F上的線性空間。定義變換
為
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稱為恆等變換或單位變換;定義變換
為
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稱為零變換,它們都是線性變換。
數乘變換的概念
設V是數域F上的線性空間,k∈F,定義變換
為
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稱為數乘變換,數乘變換是線性變換,故線性變換的性質也是數乘變換的性質,參見線性變換。顯然當k=1數乘變換即為恆等變換,k=0數乘變換即為零變換。
相關性質
(1)設
是V的一個線性變換,則:
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因為
(0)=
(0·α)=0
(α)=0,
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(2)線性變換
保持向量的線性組合和線性關係式不變,即
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若β是α1,α2,…,αs的線性組合:
β=k1α1+k2α2+…+ksαs,
則有
(β)=k1
(α1)+k2
(α2)+…+ks
(αs),
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同樣若對於α1,α2,…,αs,有:
k1α1+k2α2+…+ksαs=0,
則有:
k1
(α1)+k2
(α2)+…+ks
(αs)=0。
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(3)線性變換把線性相關的向量組變為線性相關的向量組。
注 線性變換可能把線性無關的向量組變為線性相關的向量組,譬如零變換。