线性代数 Ch03：n维向量
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上一篇：线性代数 Ch02：矩阵 · 下一篇：线性代数 Ch04：线性方程组本文是“线性代数”系列的第 03 章，主题为“n维向量”。
向量内积 \((\boldsymbol{\alpha},\boldsymbol{\beta})=\boldsymbol{\alpha}^T\boldsymbol{\beta}=\boldsymbol{\beta}^T\boldsymbol{\alpha}=a_1b_1+a_2b_2+\cdots+a_nb_n\)
(1) 向量\(\boldsymbol{\alpha}=(a_1,a_2,\cdots,a_n)^T\)的长度
\[ ||\boldsymbol{\alpha}||=\sqrt{\boldsymbol{\alpha}^T\boldsymbol{\alpha}}=\sqrt{a_1^2+a_2^2+\cdots+a_n^2} \](2) \(\boldsymbol{\alpha}^T\boldsymbol{\alpha}=0\Leftrightarrow a_1^2+a_2^2+\cdots+a_n^2=0\Leftrightarrow\boldsymbol{\alpha}=\boldsymbol{0}\).
(3) \((\boldsymbol{\alpha},\boldsymbol{\beta})=0\Leftrightarrow\boldsymbol{\alpha}\perp\boldsymbol{\beta}\).
向量\(\boldsymbol{\beta}\)可由向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)线性表出
\(\Leftrightarrow k_1\boldsymbol{\alpha}_1+k_2\boldsymbol{\alpha}_2+\cdots+k_s\boldsymbol{\alpha}_s=\boldsymbol{\beta}\)
\(\Leftrightarrow\)非齐次线性方程组\[ [\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s]\left[ \begin{array}{c} x_1\\ x_2\\ \vdots \\ x_s \end{array}\right] =\boldsymbol{\beta} \]有解
\(\Leftrightarrow r[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s]=r[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s,\boldsymbol{\beta}]\).
向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)线性相关
\(\Leftrightarrow k_1\boldsymbol{\alpha}_1+k_2\boldsymbol{\alpha}_2+\cdots+k_s\boldsymbol{\alpha}_s=0,k_i\)不全为零
\(\Leftrightarrow\)齐次线性方程组\[ [\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s]\left[ \begin{array}{c} x_1\\ x_2\\ \vdots \\ x_s \end{array}\right] =\boldsymbol{0} \]有非零解
\(\Leftrightarrow r[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s]<s\).
推论:
(1) \(n\)个\(n\)维向量线性相关\(\Leftrightarrow|\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n|=0\).
(2) \(n+1\)个\(n\)维向量一定线性相关.
线性相关和线性表出的关系
(1) 向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)相关\(\Leftrightarrow\)存在一个向量可由其余\(s-1\)个向量表出.
(2) 向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)无关, \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s,\boldsymbol{\beta}\)相关\(\Rightarrow\boldsymbol{\beta}\)可由\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)唯一表出.
向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)可由向量组\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t\)线性表出
\(\Leftrightarrow\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)中的每个向量都可由\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t\)中的向量线性表出
若两个向量组可以互相线性表出, 则向量组等价.
(1) 部分组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_r\)相关\(\Rightarrow\)整体组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_r,\cdots,\boldsymbol{\alpha}_s\)相关.
“子集合相关, 整体必相关”.
(2) \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_m\)无关\(\Rightarrow\)延伸组\(\tilde{\boldsymbol{\alpha}}_1,\tilde{\boldsymbol{\alpha}}_2,\cdots,\tilde{\boldsymbol{\alpha}}_m\)无关.
“低维无关, 高维必无关”.
(3) \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)可由\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t\)表出, 且\(s>t\Rightarrow\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)相关.
“多由少表出, 则多必相关”.
推论: \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)无关, 且可由\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t\)表出\(\Rightarrow s\leqslant t\).
(四) \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s\)可由\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t\)表出\(\Rightarrow r(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_s)\leqslant r(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_t)\).
推论:
(a) 向量组(I), (II)等价\(\Rightarrow r(\mathrm{I})=r(\mathrm{II})\).
(b) 由向量组秩相等一般不能得到向量组等价, 还需增加以下条件:
若\(r(\mathrm{I})=r(\mathrm{II})\), 且(I)可由(II)线性表出\(\Rightarrow\)向量组等价.
若\(r(\mathrm{I})=r(\mathrm{II})=r(\mathrm{I},\mathrm{II})\Rightarrow\)向量组等价.
已知向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3\)线性无关, 若\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3\)可由\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3\)表出, 设
\[ [\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3]=[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3]\boldsymbol{C} \]则\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\boldsymbol{\beta}_3\)线性无关的充要条件是\(|\boldsymbol{C}|\neq0\).
\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_m\)为向量空间\(V\)的一个基, 则
(1) \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_m\)线性无关;
(2) \(V\)中任意向量\(\boldsymbol{\beta}\)均可由向量组\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_m\)线性表出, 即
\[ x_1\boldsymbol{\alpha}_1+x_2\boldsymbol{\alpha}_2+\cdots+x_m\boldsymbol{\alpha}_m=\boldsymbol{\beta} \]向量空间\(V\)的维数\(\dim V=m\), \(\boldsymbol{\beta}\)的系数\(x_1,x_2,\cdots,x_m\)称为\(\boldsymbol{\beta}\)在基底\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_m\)下的坐标.
规范正交基\(\boldsymbol{e}_1,\boldsymbol{e}_2,\cdots,\boldsymbol{e}_n\)满足
\[ (\boldsymbol{e}_i,\boldsymbol{e}_j)=\begin{cases} 1 & i=j\\ 0 & i\neq j \end{cases} \]
\(n\)维向量空间中的两组基(I) \(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n\), (II) \(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n\), 若有
\[ [\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n]=[\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n]\boldsymbol{C} \]其中
\[ \boldsymbol{C}=\left[ \begin{array}{cccc} c_{11} & c_{12} & \cdots & c_{1n}\\ c_{21} & c_{22} & \cdots & c_{2n}\\ \vdots & \vdots & & \vdots\\ c_{n1} & c_{n2} & \cdots & c_{nn}\\ \end{array}\right] \]则称\(\boldsymbol{C}\)为由基\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n\)到基\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n\)的过渡矩阵.
(1) 过渡矩阵\(\boldsymbol{C}\)为可逆矩阵.
(2) 若\(\gamma\)在基\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n\)下的坐标为\(x_1,x_2,\cdots,x_n\), 在基\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n\)下的坐标为\(y_1,y_2,\cdots,y_n\), 则坐标变换公式
\[ \left[ \begin{array}{c} x_1\\ x_2\\ \vdots\\ x_n\\ \end{array}\right] =\boldsymbol{C}\left[ \begin{array}{c} y_1\\ y_2\\ \vdots\\ y_n\\ \end{array}\right] \]
\[ \boldsymbol{x}=\boldsymbol{C}\boldsymbol{y} \]其中\(\boldsymbol{C}\)为由基\(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2,\cdots,\boldsymbol{\alpha}_n\)到基\(\boldsymbol{\beta}_1,\boldsymbol{\beta}_2,\cdots,\boldsymbol{\beta}_n\)的过渡矩阵.
(3) \(\boldsymbol{e}_1,\boldsymbol{e}_2,\cdots,\boldsymbol{e}_n\)为规范正交基, 设
\[ [\boldsymbol{\varepsilon}_1,\boldsymbol{\varepsilon}_2,\cdots,\boldsymbol{\varepsilon}_n]=[\boldsymbol{e}_1,\boldsymbol{e}_2,\cdots,\boldsymbol{e}_n]\boldsymbol{C} \]则\(\boldsymbol{\varepsilon}_1,\boldsymbol{\varepsilon}_2,\cdots,\boldsymbol{\varepsilon}_n\)是规范正交基\(\Leftrightarrow \boldsymbol{C}\)为正交矩阵.
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