【柯】代数学引论 第2章 §1.行和列的向量空间

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\(Page.55\\1.\ 证明其逆否命题\\\quad 对\forall i,j\leqslant k,若\exists X'\in V_{i}\cap V_{j},且X'\not\equiv 0\\\quad 则X=X_{1}+X_{2}=(X_{1}+nX')+(X_{2}-nX')\\\quad 故原命题得证\)?
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\(2.\ 不一定永久成立\\\quad e.g.V_{1}=<(1,0,1)>,V_{2}=<(0,1,-1)>,V=<(1,1,0)>\\\quad 在V_{1}\subset V的特殊情况下，这一关系式成立\)
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\(3.\ 不是唯一的\\\quad e.g.设U=<(0,1)>,V=<(0,1),(1,0)>,则W=<(1,1)>或<(1,2)>或<···>\\\quad 与集合论概念下的补集的差别至此就显而易见了\)
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\(4.\ 若X_{1}与X_{2}是线性相关的,则存在不全为零的\alpha _{1},\alpha _{2}\\\quad s.t.\ \alpha _{1}X_{1}+\alpha _{2}X_{2}=0\Rightarrow \left\{\begin{array}{l}\alpha _{1}+3\alpha _{2}=0\\2\alpha _{1}+2\alpha _{2}=0\\3\alpha _{1}+\alpha _{2}=0\end{array}\right.\Rightarrow \left\{\begin{array}{l}\alpha _{1}=0\\\alpha _{2}=0\end{array}\right.\quad 矛盾\\\quad 故X_{1}与X_{2}是线性无关的\\~\\\quad \alpha _{1}X_{1}+\alpha _{2}X_{2}=(-5,2,9)\Rightarrow \left\{\begin{array}{l}\alpha _{1}+3\alpha _{2}=-5\\2\alpha _{1}+2\alpha _{2}=2\\3\alpha _{1}+\alpha _{2}=9\end{array}\right.\Rightarrow \left\{\begin{array}{l}\alpha _{1}=4\\\alpha _{2}=-3\end{array}\right.\\~\\\quad 设V在\mathbb{R}^{3}下的补X_{3}=(1,1,2),易证X_{1},X_{2},X_{3}线性无关\)
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\(5.\ \Rightarrow :\\\quad 若\mathbb{R}^{n}中的向量组X_{1},···,X_{n}张成\mathbb{R}^{n}\\\quad 则由引理2(P.54)可知X_{1},···,X_{n}线性无关\\~\\\quad \Leftarrow :\\\quad 若X_{1},···,X_{n}线性相关,即存在\alpha _{1}X_{1}+\alpha _{2}X_{2}=0\\\quad \alpha _{1}X_{1}+\alpha _{2}X_{2}+···+\alpha _{n}X_{n}=X\Rightarrow \beta _{2}X_{2}+\alpha _{3}X_{3}+···+\alpha _{n}X_{n}=X\in \mathbb{R}^{n},由引理(2)可知,X_{1},···,X_{n}不可能张成\mathbb{R}^{n}\)
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\(6.\ 见P.54 定理2\)
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\(7.\ 设U=,V=\\\quad 若U\cap V=0,则对X\in U+V,有唯一的\alpha _{i},s.t.X=\alpha _{1}X_{1}+···+\alpha _{n}X_{n}+\alpha _{n+1}Y_{1}+···+\alpha _{n+m}Y_{m}\\\quad 即X_{1},X_{2},···,X_{n},Y_{1},Y_{2},···,Y_{m}为U+V的一组基,故dim(U+V)=dim(U)+dim(V)\)
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\(8.\ 线性无关,秩为3\)