1. 向量的运算
1. 向量的定义
由 \(n\) 个数 \({a}_{1},{a}_{2},\cdots ,{a}_{n}\) 组成的有序数组称为 \(n\) 维向量,简称向量\[\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right)\]称为 \(n\) 维行向量, \({a}_{i}\) 称为 \(\boldsymbol{\alpha }\) 的第 \(i\) 个分量\[\boldsymbol{\beta } = \left\lbrack \begin{matrix} {b}_{1} \\ {b}_{2} \\ \cdots \\ {b}_{n} \end{matrix}\right\rbrack = {\left( {b}_{1},{b}_{2},\cdots ,{b}_{n}\right) }^{T}\]称为 \(n\) 维列向量
由定义可看出, \(n\) 维行 (列) 向量就是 \(1 \times n\left( {n \times 1}\right)\) 矩阵. 本书约定,所讨论的向量未指明是行向量还是列向量时, 都当作列向量
分量全为 0 的向量称为零向量; 设 \(\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right)\) ,称 \(- \boldsymbol{\alpha } = \left( {-{a}_{1}, - {a}_{2},\cdots , - {a}_{n}}\right)\) 为 \(\boldsymbol{\alpha }\) 的负向量
2. 向量的运算
(1)向量的相等 设 \(\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right) ,\boldsymbol{\beta } = \left( {{b}_{1},{b}_{2},\cdots ,{b}_{n}}\right)\) ,如果 \({a}_{i} = {b}_{i}\left( {i = 1,2,\cdots ,n}\right)\) ,则称向量 \(\boldsymbol{\alpha }\) 与 \(\boldsymbol{\beta }\) 相等
(2)向量的加减 设 \(\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right) ,\boldsymbol{\beta } = \left( {{b}_{1},{b}_{2},\cdots ,{b}_{n}}\right)\) ,则 \(\boldsymbol{\alpha } \pm \boldsymbol{\beta } = \left( {{a}_{1} \pm {b}_{1},{a}_{2} \pm {b}_{2},\cdots ,{a}_{n}}\right.\) \(\left. {\pm {b}_{n}}\right)\)
(3)向量的数乘 设 \(\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right) ,k\) 为常数,则 \(k\boldsymbol{\alpha } = \left( {k{a}_{1},k{a}_{2},\cdots ,k{a}_{n}}\right)\)
3. 向量的运算规律
设 \(\boldsymbol{\alpha },\boldsymbol{\beta },\boldsymbol{\gamma }\) 均为 \(n\) 维向量, \(\lambda ,\mu\) 为实数,则
(1)\(\boldsymbol{\alpha } + \boldsymbol{\beta } = \boldsymbol{\beta } + \boldsymbol{\alpha }\)
(2)\(\left( {\boldsymbol{\alpha } + \boldsymbol{\beta }}\right) + \gamma = \boldsymbol{\alpha } + \left( {\boldsymbol{\beta } + \gamma }\right)\)
(3)\(\boldsymbol{\alpha } + \boldsymbol{0} = \boldsymbol{\alpha }\)
(4)\(\boldsymbol{\alpha } + \left( {-\boldsymbol{\alpha }}\right) = \boldsymbol{0}\)
(5)\(1\boldsymbol{\alpha } = \boldsymbol{\alpha }\)
(6)\(\lambda \left( {\mu \boldsymbol{\alpha }}\right) = \left( {\lambda \mu }\right) \boldsymbol{\alpha }\)
(7)\(\lambda \left( {\boldsymbol{\alpha } + \boldsymbol{\beta }}\right) = \lambda \boldsymbol{\alpha } + \lambda \boldsymbol{\beta }\)
(8)\(\left( {\lambda + \mu }\right) \boldsymbol{\alpha } = \lambda \boldsymbol{\alpha } + \mu \boldsymbol{\alpha }\)
2. 向量间的线性关系
1. 基本概念
(1)线性表示 对于向量 \(\boldsymbol{\beta },{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) ,如果存在一组数 \({k}_{1},{k}_{2},\cdots ,{k}_{m}\) ,使得\[\boldsymbol{\beta } = {k}_{1}{\boldsymbol{\alpha }}_{1} + {k}_{2}{\boldsymbol{\alpha }}_{2} + \cdots + {k}_{m}{\boldsymbol{\alpha }}_{m}\]成立,则称 \(\boldsymbol{\beta }\) 是 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的线性组合,或称 \(\boldsymbol{\beta }\) 可由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示
(2)线性相关与线性无关 设 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 为一组向量,如果存在一组不全为零的数 \({k}_{1}\) , \({k}_{2},\cdots {k}_{m}\) ,使得\[{k}_{1}{\boldsymbol{\alpha }}_{1} + {k}_{2}{\boldsymbol{\alpha }}_{2} + \cdots + {k}_{m}{\boldsymbol{\alpha }}_{m} = \boldsymbol{0}\]成立,则称向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性相关; 当且仅当 \({k}_{1} = {k}_{2} = \cdots = {k}_{m} = 0\) 时等式成立,则称向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关
2. 常用结论
设 \({\boldsymbol{\alpha }}_{1} = {\left( {a}_{11},{a}_{12},\cdots ,{a}_{1n}\right) }^{T},{\boldsymbol{\alpha }}_{2} = {\left( {a}_{21},{a}_{22},\cdots ,{a}_{2n}\right) }^{T},\cdots ,{\boldsymbol{\alpha }}_{m} = {\left( {a}_{m1},{a}_{m2},\cdots ,{a}_{mn}\right) }^{T}\) ,\(\boldsymbol{\beta } = {\left( {b}_{1},{b}_{2},\cdots ,{b}_{n}\right) }^{T}\) ,这里 \(m \leq n\)
(1)\(\boldsymbol{\beta }\) 可由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示的充分必要条件是线性方程组 \({x}_{1}{\boldsymbol{\alpha }}_{1} + {x}_{2}{\boldsymbol{\alpha }}_{2} + \cdots + {x}_{m}{\boldsymbol{\alpha }}_{m} =\) \(\boldsymbol{\beta }\) 有解,即下列线性方程组有解\[\left\{ \begin{array}{l} {a}_{11}{x}_{1} + {a}_{21}{x}_{2} + \cdots + {a}_{m1}{x}_{m} = {b}_{1}, \\ {a}_{12}{x}_{1} + {a}_{22}{x}_{2} + \cdots + {a}_{m2}{x}_{m} = {b}_{2}, \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ {a}_{1n}{x}_{1} + {a}_{2n}{x}_{2} + \cdots + {a}_{mn}{x}_{m} = {b}_{n}. \end{array}\right.\](2)① 令 \(\boldsymbol{A} = \left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right) ,\boldsymbol{B} = \left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m},\boldsymbol{\beta }}\right)\) ,则 \(\boldsymbol{\beta }\) 可由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示的充分必要条件是以 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 为列向量的矩阵和以 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m},\boldsymbol{\beta }\) 为列向量的矩阵有相同的秩,即 \(r\left( \boldsymbol{A}\right) = r\left( \boldsymbol{B}\right)\)
②\(\boldsymbol{\beta }\) 可由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 唯一线性表示的充分必要条件是 \(r\left( \boldsymbol{A}\right) = r\left( \boldsymbol{B}\right) = m\)
③\(\boldsymbol{\beta }\) 不能由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示的充分必要条件是 \(r\left( \boldsymbol{A}\right) < r\left( \boldsymbol{B}\right)\)
(3)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性相关的充分必要条件是齐次线性方程组\[\left\{ \begin{array}{l} {a}_{11}{x}_{1} + {a}_{21}{x}_{2} + \cdots + {a}_{m1}{x}_{m} = 0, \\ {a}_{12}{x}_{1} + {a}_{22}{x}_{2} + \cdots + {a}_{m2}{x}_{m} = 0, \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ {a}_{1m}{x}_{1} + {a}_{2m}{x}_{2} + \cdots + {a}_{mm}{x}_{m} = 0 \end{array}\right.\]有非零解,且当 \(m = n\) 时,其线性相关的充要条件是\[\left| \boldsymbol{A}\right| = \left| \begin{matrix} {a}_{11} & {a}_{12} & \cdots & {a}_{1n} \\ {a}_{21} & {a}_{22} & \cdots & {a}_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ {a}_{n1} & {a}_{n2} & \cdots & {a}_{nn} \end{matrix}\right| = 0\](4)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关的充分必要条件是齐次线性方程组\[\left\{ \begin{array}{l} {a}_{11}{x}_{1} + {a}_{21}{x}_{2} + \cdots + {a}_{m1}{x}_{m} = 0, \\ {a}_{12}{x}_{1} + {a}_{22}{x}_{2} + \cdots + {a}_{m2}{x}_{m} = 0, \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ {a}_{1n}{x}_{1} + {a}_{2n}{x}_{2} + \cdots + {a}_{mn}{x}_{m} = 0 \end{array}\right.\]只有零解,且当 \(m = n\) 时,其线性无关的充要条件是\[\left| \boldsymbol{A}\right| = \left| \begin{matrix} {a}_{11} & {a}_{12} & \cdots & {a}_{1n} \\ {a}_{21} & {a}_{22} & \cdots & {a}_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ {a}_{m1} & {a}_{m2} & \cdots & {a}_{mn} \end{matrix}\right| \neq 0\](5)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性相关的充要条件是以 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 为列向量的矩阵的秩小于向量个数 \(m\)
(6)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关的充要条件是以 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 为列向量的矩阵的秩等于向量个数 \(m\)
(7)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\left( {m \geq 2}\right)\) 线性相关的充要条件是向量组至少有一个向量是其余向量的线性组合; 向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\left( {m \geq 2}\right)\) 线性无关的充要条件是向量组中每一个向量都不能由其余向量线性表示
(8)如果向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关,而向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m},\boldsymbol{\beta }\) 线性相关,则 \(\boldsymbol{\beta }\) 可以由 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示,且表达式唯一
(9)如果向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 可以由向量组 \({\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{t}\) 线性表示,并且 \(m > t\) ,则向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性相关; 或者说,如果向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关,并且可以由 \({\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{t}\) 线性表示,则 \(m \leq t\)
(10)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 中,如果有一个部分组线性相关,则整个向量组线性相关; 如果整个向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关,则其任一部分组也一定线性无关
(11)设 \(r\) 维向量组 \({\boldsymbol{\alpha }}_{i} = \left( {{a}_{i1},{a}_{i2},\cdots ,{a}_{ir}}\right) \left( {i = 1,2,\cdots ,m}\right)\) 线性无关,则在每个向量上再添加 \(n - r\) 个分量所得到的 \(n\) 维向量组 \({\boldsymbol{\alpha }}_{i}^{\prime } = \left( {{a}_{i1},{a}_{i2},\cdots ,{a}_{ir},{a}_{{ir} + 1},\cdots ,{a}_{in}}\right) \left( {i = 1,2,\cdots ,m}\right)\) 也线性无关
(12)\(n + 1\) 个 \(n\) 维向量必线性相关
(13)一个零向量线性相关；含有零向量的向量组必线性相关；一个非零向量线性无关；两个非零向量线性相关的充要条件是对应分量成比例
(14) 设 \({\varepsilon }_{1} = \left( {1,0,\cdots ,0}\right) ,{\varepsilon }_{2} = \left( {0,1,\cdots ,0}\right) ,\cdots ,{\varepsilon }_{n} = \left( {0,0,\cdots ,1}\right)\) ,称 \({\varepsilon }_{1},{\varepsilon }_{2},\cdots ,{\varepsilon }_{n}\) 为 \(n\) 维单位向量组, 且
①\({\varepsilon }_{1},{\varepsilon }_{2},\cdots ,{\varepsilon }_{n}\) 线性无关
②任意 \(n\) 维向量 \(\boldsymbol{\alpha } = \left( {{a}_{1},{a}_{2},\cdots ,{a}_{n}}\right)\) 都可由 \({\boldsymbol{\varepsilon }}_{1},{\boldsymbol{\varepsilon }}_{2},\cdots ,{\boldsymbol{\varepsilon }}_{n}\) 线性表示,即 \(\boldsymbol{\alpha } = {a}_{1}{\boldsymbol{\varepsilon }}_{1} + {a}_{2}{\boldsymbol{\varepsilon }}_{2} + \cdots +\) \({a}_{n}{\varepsilon }_{n}\)
(15)初等行变换不改变矩阵的列向量组之间的线性关系; 初等列变换不改变矩阵的行向量组之间的线性关系
3. 向量组的极大线性无关组和秩
1. 极大无关组
设向量组 \({\boldsymbol{\alpha }}_{i1},{\boldsymbol{\alpha }}_{i2},\cdots ,{\boldsymbol{\alpha }}_{ir}\) 为向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的一个部分组,且满足
(1)\({\boldsymbol{\alpha }}_{i1},{\boldsymbol{\alpha }}_{i2},\cdots ,{\boldsymbol{\alpha }}_{ir}\) 线性无关
(2)向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 中任一向量均可由 \({\boldsymbol{\alpha }}_{i1},{\boldsymbol{\alpha }}_{i2},\cdots ,{\boldsymbol{\alpha }}_{ir}\) 线性表示
则称向量组 \({\boldsymbol{\alpha }}_{i1},{\boldsymbol{\alpha }}_{i2},\cdots ,{\boldsymbol{\alpha }}_{ir}\) 为向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的一个极大线性无关组,简称极大无关组
2. 向量组的秩 向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的极大无关组中所含向量的个数为该向量组的秩,记为 \(r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right)\)
如果一个向量组仅含有零向量, 则规定它的秩为零
3. 向量组的秩的性质
(1)若 \(r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right) = r\) ,则
①\({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的任何含有多于 \(r\) 个向量的部分组一定线性相关
②\({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 的任何含 \(r\) 个向量的线性无关部分组一定是极大无关组
(2)\(r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right) \leq m\) ,且 \(r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right) = m \Leftrightarrow {\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性无关
(3)向量 \(\boldsymbol{\beta }\) 可用 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}\) 线性表示 \(\Leftrightarrow r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m},\boldsymbol{\beta }}\right) = r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{m}}\right)\)
(4)若 \({\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{t}\) 可用 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{s}\) 线性表示,则 \(r\left( {{\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{t}}\right) \leq r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{s}}\right)\)
(5)设 \(\boldsymbol{A}\) 是一个 \(m \times n\) 矩阵,记 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{n}\) 是 \(\boldsymbol{A}\) 的列向量组 \(\left( {m\text{ 维 }}\right) ,{\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{m}\) 是 \(\boldsymbol{A}\) 的行向量组 ( \(n\) 维),则 \(r\left( \boldsymbol{A}\right) = r\left( {{\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{n}}\right) = r\left( {{\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{m}}\right)\)
4. 向量组的等价
两个向量组能够相互线性表示, 则称这两个向量组等价
向量组等价的结论:
(1)任一向量组和它的极大无关组等价
(2)向量组的任意两个极大无关组等价
(3)两个等价的线性无关的向量组所含向量的个数相同
(4) 两个向量组等价的充要条件是它们的极大无关组等价
(5)等价的两个向量组有相同的秩
4. 向量的内积与向量空间
1. 向量的内积
给定 \({\boldsymbol{R}}^{n}\) 中向量\[\boldsymbol{\alpha } = {\left( {a}_{1},{a}_{2},\cdots ,{a}_{n}\right) }^{T},\;\boldsymbol{\beta } = {\left( {b}_{1},{b}_{2},\cdots ,{b}_{n}\right) }^{T}\]则称 \(\mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{b}_{i}\) 为向量 \(\boldsymbol{\alpha }\) 与 \(\boldsymbol{\beta }\) 的内积,记为 \(\left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right)\) ,即 \(\left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right) = {\boldsymbol{\alpha }}^{T}\boldsymbol{\beta } = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{b}_{i}\)
内积具有下列性质:
(1)\(\left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right) = \left( {\boldsymbol{\beta },\boldsymbol{\alpha }}\right)\)
(2)\(\left( {k\boldsymbol{\alpha },\boldsymbol{\beta }}\right) = k\left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right)\)
(3)\(\left( {\boldsymbol{\alpha } + \boldsymbol{\beta },\gamma }\right) = \left( {\boldsymbol{\alpha },\gamma }\right) + \left( {\boldsymbol{\beta },\gamma }\right)\)
(4)\(\left( {\boldsymbol{\alpha },\boldsymbol{\alpha }}\right) \geq 0\) ,当且仅当 \(\boldsymbol{\alpha } = \boldsymbol{0}\) 时,等号成立
2. 向量的范数
设 \(\boldsymbol{\alpha }\) 为 \({\boldsymbol{R}}^{n}\) 中任意向量,将非负实数 \(\sqrt{\left( \boldsymbol{\alpha },\boldsymbol{\alpha }\right) }\) 定义为 \(\boldsymbol{\alpha }\) 的长度,记为 \(\parallel \boldsymbol{\alpha }\parallel\) ,即若 \(\boldsymbol{\alpha } = {\left( {a}_{1},{a}_{2},\cdots ,{a}_{n}\right) }^{T}\) ,则有\[\parallel \boldsymbol{\alpha }\parallel = \sqrt{\left( {a}_{1}^{2} + {a}_{2}^{2} + \cdots + {a}_{n}^{2}\right) }\]向量的长度也称为向量的范数或模
向量范数具有下列性质:
(1)\(\parallel \boldsymbol{\alpha }\parallel \geq 0\) ,当且仅当 \(\boldsymbol{\alpha } = \boldsymbol{0}\) 时,等号成立
(2)对于任意向量 \(\boldsymbol{\alpha }\) 和任意实数 \(k\) ,都有 \(\parallel k\boldsymbol{\alpha }\parallel = \left| k\right| \parallel \boldsymbol{\alpha }\parallel\)
(3)对于任意 \(n\) 维向量 \(\boldsymbol{\alpha }\) 和 \(\boldsymbol{\beta }\) ,有 \(\left| \left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right) \right| = \left| {{\boldsymbol{\alpha }}^{T}\boldsymbol{\beta }}\right| \leq \parallel \boldsymbol{\alpha }\parallel \parallel \boldsymbol{\beta }\parallel\)
3. 向量的正交
如果向量 \(\boldsymbol{\alpha }\) 和 \(\boldsymbol{\beta }\) 的内积等于零,即 \(\left( {\boldsymbol{\alpha },\boldsymbol{\beta }}\right) = 0\) ,则称 \(\boldsymbol{\alpha }\) 和 \(\boldsymbol{\beta }\) 相互正交
如果非零向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{s}\) 中向量两两正交,即 \(\left( {{\boldsymbol{\alpha }}_{i},{\boldsymbol{\alpha }}_{j}}\right) = 0\left( {i \neq j,i,j = 1,2,\cdots ,s}\right)\) ,则称该向量组为正交向量组
正交向量具有下列性质:
(1)零向量与任何向量正交
(2)与自己正交的向量只有零向量
(3)正交向量组是线性无关的
(4)对任意向量 \(\boldsymbol{\alpha }\) 和 \(\boldsymbol{\beta }\) ,有三角不等式\[\parallel \boldsymbol{\alpha } + \boldsymbol{\beta }\parallel \leq \parallel \boldsymbol{\alpha }\parallel + \parallel \boldsymbol{\beta }\parallel\]当且仅当 \(\boldsymbol{\alpha }\) 与 \(\boldsymbol{\beta }\) 相互正交时,有 \(\parallel \boldsymbol{\alpha } + \boldsymbol{\beta }{\parallel }^{2} = \parallel \boldsymbol{\alpha }{\parallel }^{2} + \parallel \boldsymbol{\beta }{\parallel }^{2}\)
4. 向量空间
设 \(\boldsymbol{V}\) 是实数域 \(\boldsymbol{R}\) 上的 \(n\) 维向量组成的集合,如果 \(\boldsymbol{V}\) 关于向量的加法和数乘是封闭的, 即
若 \(\alpha \in V,\beta \in V\) ,则 \(\alpha + \beta \in V\) ; 若 \(\alpha \in V,k \in R\) ,则 \({k\alpha } \in V\) ,则称 \(V\) 是实数域 \(R\) 上的向量空间
显然,实数域 \(\boldsymbol{R}\) 上的 \(n\) 维向量的全体构成一个向量空间,记为 \({\boldsymbol{R}}^{n}\)
5. 基与坐标 在向量空间 \({\boldsymbol{R}}^{n}\) 中, \(n\) 个线性无关的向量 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 称为 \({\boldsymbol{R}}^{n}\) 的一组基. 若 \(\boldsymbol{\alpha } \in {\boldsymbol{R}}^{n}\) 为任一向量,且\[\boldsymbol{\alpha } = {a}_{1}{\boldsymbol{\xi }}_{1} + {a}_{2}{\boldsymbol{\xi }}_{2} + \cdots + {a}_{n}{\boldsymbol{\xi }}_{n}\]则称 \({a}_{1},{a}_{2},\cdots ,{a}_{n}\) 为 \(\boldsymbol{\alpha }\) 关于基 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 的坐标,记作 \({\left( {a}_{1},{a}_{2},\cdots ,{a}_{n}\right) }^{T}\)
6. 基变换与坐标变换
设 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 和 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 是 \({\boldsymbol{R}}^{n}\) 的两组基,且有\[\left( {{\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}}\right) = \left( {{\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}}\right) \left\lbrack \begin{matrix} {a}_{11} & {a}_{12} & \cdots & {a}_{1n} \\ {a}_{21} & {a}_{22} & \cdots & {a}_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ {a}_{n1} & {a}_{n2} & \cdots & {a}_{nn} \end{matrix}\right\rbrack = \left( {{\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}}\right) \boldsymbol{A}\]称 \(\boldsymbol{A}\) 为由基 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 到基 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 的过渡矩阵,两个基之间的过渡矩阵是可逆矩阵
设 \(\boldsymbol{\alpha } \in {\boldsymbol{R}}^{n}\) 在基 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 和基 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 下的坐标分别为\[{\left( {x}_{1},{x}_{2},\cdots ,{x}_{n}\right) }^{T}\text{ 与 }{\left( {y}_{1},{y}_{2},\cdots ,{y}_{n}\right) }^{T}\]则有\[\left\lbrack \begin{matrix} {x}_{1} \\ {x}_{2} \\ \cdots \\ {x}_{n} \end{matrix}\right\rbrack = \boldsymbol{A}\left\lbrack \begin{matrix} {y}_{1} \\ {y}_{2} \\ \cdots \\ {y}_{n} \end{matrix}\right\rbrack \text{ 或 }\left\lbrack \begin{matrix} {y}_{1} \\ {y}_{2} \\ \cdots \\ {y}_{n} \end{matrix}\right\rbrack = {\boldsymbol{A}}^{-1}\left\lbrack \begin{matrix} {x}_{1} \\ {x}_{2} \\ \cdots \\ {x}_{n} \end{matrix}\right\rbrack\]称其为坐标变换公式
7. \({\boldsymbol{R}}^{n}\) 的标准正交基
向量空间 \({\boldsymbol{R}}^{n}\) 中 \(n\) 个向量 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 满足
(1)两两正交,即 \({\boldsymbol{\eta }}_{i}^{T}{\boldsymbol{\eta }}_{j} = 0,i \neq j,i,j = 1,2,\cdots ,n\)
(2)都是单位向量,即 \(\begin{Vmatrix}{\boldsymbol{\eta }}_{i}\end{Vmatrix} = 1,i = 1,2,\cdots ,n\)
则称 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 为 \({\boldsymbol{R}}^{n}\) 的一组标准正交基
8. 标准正交基的求法
(1)施密特正交化方法 给定一线性无关向量组 \({\boldsymbol{\alpha }}_{1},{\boldsymbol{\alpha }}_{2},\cdots ,{\boldsymbol{\alpha }}_{s}\) ,由其生成等价的 \(s\) 个向量的正交向量组 \({\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{2},\cdots ,{\boldsymbol{\beta }}_{s}\) 的公式如下:
\({\boldsymbol{\beta }}_{1} = {\boldsymbol{\alpha }}_{1}\)
\(\displaystyle{\boldsymbol{\beta }}_{2} = {\boldsymbol{\alpha }}_{2} - \frac{\left( {\boldsymbol{\alpha }}_{2},{\boldsymbol{\beta }}_{1}\right) }{\left( {\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{1}\right) }{\boldsymbol{\beta }}_{1}\)
\(\displaystyle{\boldsymbol{\beta }}_{3} = {\boldsymbol{\alpha }}_{3} - \frac{\left( {\boldsymbol{\alpha }}_{3},{\boldsymbol{\beta }}_{1}\right) }{\left( {\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{1}\right) }{\boldsymbol{\beta }}_{1} - \frac{\left( {\boldsymbol{\alpha }}_{3},{\boldsymbol{\beta }}_{2}\right) }{\left( {\boldsymbol{\beta }}_{2},{\boldsymbol{\beta }}_{2}\right) }{\boldsymbol{\beta }}_{2}\)
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\(\displaystyle{\boldsymbol{\beta }}_{s} = {\boldsymbol{\alpha }}_{s} - \frac{\left( {\boldsymbol{\alpha }}_{s},{\boldsymbol{\beta }}_{1}\right) }{\left( {\boldsymbol{\beta }}_{1},{\boldsymbol{\beta }}_{1}\right) }{\boldsymbol{\beta }}_{1} - \frac{\left( {\boldsymbol{\alpha }}_{s},{\boldsymbol{\beta }}_{2}\right) }{\left( {\boldsymbol{\beta }}_{2},{\boldsymbol{\beta }}_{2}\right) }{\boldsymbol{\beta }}_{2} - \cdots - \frac{\left( {\boldsymbol{\alpha }}_{s},{\boldsymbol{\beta }}_{s - 1}\right) }{\left( {\boldsymbol{\beta }}_{s - 1},{\boldsymbol{\beta }}_{s - 1}\right) }{\boldsymbol{\beta }}_{s - 1}\)
(2)给定 \({\boldsymbol{R}}^{n}\) 任意一组基,把它变为标准正交基的步骤如下:
①利用施密特正交化方法,由这组基生成有 \(n\) 个向量的正交向量组
②把正交向量组中每个向量标准化, 即单位化
这样就得到 \({\boldsymbol{R}}^{n}\) 的一组标准正交基. 这一过程称为标准正交化
9. 两组标准正交基之间的过渡矩阵
设 \({\boldsymbol{R}}^{n}\) 的两组标准正交基 \({\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}\) 和 \({\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}\) 间的过渡矩阵为 \(Q\) ,则存在下列关系\[\left( {{\boldsymbol{\xi }}_{1},{\boldsymbol{\xi }}_{2},\cdots ,{\boldsymbol{\xi }}_{n}}\right) = \left( {{\boldsymbol{\eta }}_{1},{\boldsymbol{\eta }}_{2},\cdots ,{\boldsymbol{\eta }}_{n}}\right) Q\]且 \(\boldsymbol{Q}\) 满足 \({\boldsymbol{Q}}^{T}\boldsymbol{Q} = \boldsymbol{E}\) ,即 \(\boldsymbol{Q}\) 为正交矩阵