第一讲
线性代数回顾
定理和性质
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设\(A=(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m})\),其中\(\alpha_{i}\)是一个n维列向量,那么有下面命题等价:
1.1. \(b\in L(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m})\),其中$ L(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m})$是向量组张成的线性空间(线性组合的全体)
1.2. \(AX=b\)有解
1.3. \(R(\bar{A})=R(A)\),其中\(\bar{A}\)是增广矩阵 -
设\(A=(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m})\),其中\(\alpha_{i}\)是一个n维列向量,那么有下面命题等价:
2.1. \(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m}\)线性相关
2.2. \(AX=0\)有非零解
2.3. \(R(A)<m\)
2.4. \(det(A)=0\) -
\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m}\)线性相关的充要条件是:其中至少有一个向量可由其余m - 1个向量线性表出
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若\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m}\)线性无关,\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m},\beta\)线性相关。那么\(\beta\)可以由\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{m}\)线性表出,且表出形式唯一
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矩阵经过行(列)初等变换不改变列(行)向量的线性相关性
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矩阵的 行秩=列秩=矩阵的秩
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若向量组\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{r}\)可以由向量组\(\beta_{1},\beta_{2},\beta_{3},...,\beta_{s}\)线性表出,且\(\alpha_{1},\alpha_{2},\alpha_{3},...,\alpha_{r}\)线性无关,那么\(r\leq s\).
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设\(R(A)=r<n\), \(W=\{X\in R^{n}|AX=0\}\),那么W的一组基的向量个数是n-r,即\(dim(W)=n-r\)
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设\(\eta\)为\(AX=b\)的解,\(\xi\)为\(AX=0\)的解,则\(\eta+\xi\)为\(AX=b\)的解