线代强化NO8|向量|运算|线性相关|内积正交|施密特正交化|线性表示的判定证明

基本概念

向量及其运算

1. 向量
   $$\begin{array}{rl}& \text{由}\boxed{n}\text{个实数}{a}_1,a_2,\cdots ,a_n\text{组成的}n\text{元有序数组}[a_1,a_2,\cdots ,a_n]\text{称之为}\boxed{n}\text{维行向量}，\text{如果该}\\ & \text{数组是纵向排列的：}\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right]，\text{则称其为}\boxed{n}\text{维列向量}.\\ & \text{由多个同型向量（维数相同且都为行向量或列向量）组成的集合称之为}\boxed{\text{向量组}}.\end{array}$$
2. 向量的线性运算
   $$\begin{array}{rl}& \text{假设}\boxed{\alpha }=\left[\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right],\boxed{\beta }=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]，\text{则可以定义如下运算：}\\ & \text{转置：}{\boxed{\alpha }}^T=[a_1,a_2,\cdots ,a_n]，\text{通常我们也习惯把列向量写成}\boxed{\alpha }=[a_1,a_2,\cdots ,a_n{]}^T\\ & \text{向量加法：}\boxed{\alpha }+\boxed{\beta }=[a_1+b_1,a_2+b_2,\cdots ,a_n+b_n{]}^T\\ & \text{向量数乘：}k\boxed{\alpha }=[k{a}_1,k{a}_2,\cdots ,k{a}_n{]}^T\end{array}$$

线性组合与线性表示

1. 线性组合
   $$\begin{array}{rl}& \text{设}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{是}m\text{个}n\text{维向量，}{k}_1,k_2,\cdots ,k_m\text{是}m\text{个常数，则称}\\ & k_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+\cdots +k_m{\boxed{\alpha }}_m\text{为向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{的一个线性组合.}\end{array}$$
2. 线性表示
   $$\begin{array}{rl}& \text{设}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{是}m\text{个}n\text{维向量，}\\ & \boxed{\beta }\text{是一个}n\text{维向量，如果存在常数}{k}_1,k_2,\cdots ,k_m\text{，}\\ & \text{使得}\boxed{\beta }=k_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+\cdots +k_m{\boxed{\alpha }}_m\text{，}\\ & \text{则称}\boxed{\beta }\text{为向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{的一个线性组合，}\\ & \text{或称向量}\boxed{\beta }\text{可以由向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性表示.}\end{array}$$
3. 向量组的线性表示与等价
   $$\begin{array}{rl}& \text{设有向量组}(\boxed{I}):{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{与向量组}(\boxed{II}):{\boxed{\beta }}_1,{\boxed{\beta }}_2,\cdots ,{\boxed{\beta }}_t\text{，如果向量组}(\boxed{I})\text{中的}\\ & \text{每一个向量都能由向量组}(\boxed{II})\text{线性表示，则称向量组}(\boxed{I})\text{能由向量组}(\boxed{II})\text{线性表示.如}\\ & \text{果向量组}(\boxed{I})\text{与向量组}(\boxed{II})\text{能够相互线性表示，则称向量组}(\boxed{I})\text{与向量组}(\boxed{II})\text{等价，记}\\ & \text{作向量组}(\boxed{I})\cong \text{向量组}(\boxed{II}).\end{array}$$

线性相关性

1. 线性相关
   $$\begin{array}{rl}& \text{设}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{是}m\text{个}n\text{维向量，如果存在不全为零的常数}{k}_1,k_2,\cdots ,k_m\text{，使得：}\\ & k_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+\cdots +k_m{\boxed{\alpha }}_m=\boxed{0}\text{，则称向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性相关.}\end{array}$$
2. 线性无关
   $$\begin{array}{rl}& \text{如果向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{不是线性相关的，则称该向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性无关.}\\ & \text{即：当且仅当}{k}_1,k_2,\cdots ,k_m\text{全为零时，才能使}{k}_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+\cdots +k_m{\boxed{\alpha }}_m=\boxed{0}\text{成立.}\end{array}$$

内积与正交

1. 内积
   $$\begin{array}{rl}& \text{假设}\boxed{\alpha }=[a_1,a_2,\cdots ,a_n{]}^T，\boxed{\beta }=[b_1,b_2,\cdots ,b_n{]}^T\text{，则定义}\boxed{\alpha }\text{和}\boxed{\beta }\text{的内积}\\ & (\boxed{\alpha },\boxed{\beta })={\boxed{\alpha }}^T\boxed{\beta }=\sum _{i=1}^n{a}_i{b}_i.\end{array}$$
2. 正交
   $$如果向量\boxed{\alpha }和\boxed{\beta }的内积(\boxed{\alpha },\boxed{\beta })=0，则称向量\boxed{\alpha }和\boxed{\beta }正交.$$
3. 正交向量组
   $$\begin{array}{rl}& \text{设}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{为由非零向量组成的向量组，如果其中任意两个向量都是正交的，则}\\ & \text{称该向量组为正交向量组.}\end{array}$$

重要公式与定理

常见性质

$$\begin{array}{rl}& \text{定理 1：已知向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性无关，且向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m,\boxed{\beta }\text{线性相关，则}\\ & \boxed{\beta }\text{可以由向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性表示.}\\ & \\ & \text{定理 2：向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性相关当且仅当}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{中至少有一个向量是其}\\ & \text{余}m-1\text{个向量的线性组合.}\\ & \\ & \text{定理 3：若向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性相关，则向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m,{\boxed{\alpha }}_{m+1}\text{也线性相关.}\end{array}$$
$$\begin{array}{rl}& \text{定理 4：若向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_s\text{可以由向量组}{\boxed{\beta }}_1,{\boxed{\beta }}_2,\cdots ,{\boxed{\beta }}_t\text{线性表示，且}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_s\\ & \text{线性无关，则有}s\le t.\\ & \\ & \text{定理 5：}n+1\text{个}n\text{维向量必然线性相关.}\\ & \\ & \text{定理 6：若向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{线性无关，则向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_m\text{的延伸组}\\ & \left[\begin{array}{c}{\boxed{\alpha }}_1\\ {\boxed{\beta }}_1\end{array}\right],\left[\begin{array}{c}{\boxed{\alpha }}_2\\ {\boxed{\beta }}_2\end{array}\right],\cdots ,\left[\begin{array}{c}{\boxed{\alpha }}_m\\ {\boxed{\beta }}_m\end{array}\right]\text{也线性无关.}\\ & \\ & \text{定理 7：阶梯形向量组线性无关.}\end{array}$$

与线性方程组有关的定理

$$\begin{array}{rl}& \text{定理 1：向量}\boxed{\beta }\text{可以由向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\text{线性表示}\\ & \iff r\left({\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\right)=r\left({\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n,\boxed{\beta }\right)\\ & \iff \text{线性方程组}\left[{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\boxed{\beta }\text{有解.}\\ & \\ & \text{定理 2：向量组}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\text{线性相关}\\ & \iff r\left({\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\right)<n\\ & \iff \text{齐次线性方程组}\left[{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\boxed{0}\text{有非零解.}\end{array}$$

施密特正交化

$$\begin{array}{rl}& 假设{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n线性无关，令\\ & {\boxed{\beta }}_1={\boxed{\alpha }}_1,{\boxed{\beta }}_2={\boxed{\alpha }}_2-\frac{({\boxed{\alpha }}_2,{\boxed{\beta }}_1)}{({\boxed{\beta }}_1,{\boxed{\beta }}_1)}{\boxed{\beta }}_1,\cdots ,{\boxed{\beta }}_n={\boxed{\alpha }}_n-\sum _{i=1}^{n-1}\frac{({\boxed{\alpha }}_n,{\boxed{\beta }}_i)}{({\boxed{\beta }}_i,{\boxed{\beta }}_i)}{\boxed{\beta }}_i\\ & 这样得到的{\boxed{\beta }}_1,{\boxed{\beta }}_2,\cdots ,{\boxed{\beta }}_n是和{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,\cdots ,{\boxed{\alpha }}_n等价的正交向量组.\end{array}$$

线性表示的判定与证明

$$\begin{array}{rl}& ({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3⋮\boxed{\beta })\\ & \text{唯一解：}r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3)=r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3⋮\boxed{\beta })=3\\ & \text{无解：}r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3)<r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3⋮\boxed{\beta })\\ & \text{无穷多解：}r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3)=r({\boxed{\alpha }}_1{\boxed{\alpha }}_2{\boxed{\alpha }}_3⋮\boxed{\beta })<3\end{array}$$
$$\begin{array}{rl}& \text{解：设有一组数}{k}_1,k_2,k_3\text{，满足}{k}_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+k_3{\boxed{\alpha }}_3=\boxed{\beta }\text{，增广矩阵为}\\ & \overline{A}=\left(\begin{array}{ccccc}\alpha & -2& -1& |& 1\\ 2& 1& 1& |& b\\ 10& 5& 4& |& c\end{array}\right)\to \left(\begin{array}{ccccc}2& 1& 1& |& b\\ 0& -2-\frac{\alpha }{2}& -1-\frac{\alpha }{2}& |& 1-\frac{\alpha b}{2}\\ 0& 0& 1& |& 5b-c\end{array}\right)\end{array}$$
$$\begin{array}{rl}& (1)\text{当}-2-\frac{\alpha }{2}\ne 0\text{时，}r(A)=r(\overline{A})=3\text{，方程组有唯一解，即}\boxed{\beta }\text{可由}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,{\boxed{\alpha }}_3\text{唯一线性表示.}\\ & (2)\text{当}-2-\frac{\alpha }{2}=0\text{，即}\alpha =-4\text{时，}\overline{A}=\left(\begin{array}{ccccc}2& 1& 1& |& b\\ 0& 0& 1& |& 1+2b\\ 0& 0& 0& |& 3b-c-1\end{array}\right)\\ & \text{若}3b-c-1\ne 0\text{，则}r(A)=2,r(\overline{A})=3\text{，故方程组无解，即}\boxed{\beta }\text{不可由}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,{\boxed{\alpha }}_3\text{线性表示.}\\ & \text{若}3b-c-1=0\text{，}\overline{A}=\left(\begin{array}{ccccc}2& 1& 1& |& b\\ 0& 0& 1& |& 1+2b\\ 0& 0& 0& |& 0\end{array}\right)\to \left(\begin{array}{ccccc}2& 1& 0& |& -1-b\\ 0& 0& 1& |& 1+2b\\ 0& 0& 0& |& 0\end{array}\right)\to \left(\begin{array}{ccccc}1& \frac{1}{2}& 0& |& -\frac{1+b}{2}\\ 0& 0& 1& |& 1+2b\\ 0& 0& 0& |& 0\end{array}\right)\end{array}$$
$$\begin{array}{rl}& \text{非齐次方程通解为 }k{\left(-\frac{1}{2},1,0\right)}^T+{\left(-\frac{1+b}{2},0,1+2b\right)}^T={\left(-\frac{k}{2}-\frac{1+b}{2},k,1+2b\right)}^T\\ & \text{即 }\boxed{\beta }=\left(-\frac{k}{2}-\frac{1+b}{2}\right){\boxed{\alpha }}_1+k{\boxed{\alpha }}_2+(1+2b){\boxed{\alpha }}_3\end{array}$$

$$\begin{array}{rl}& \text{【小结】}\\ & ①\text{注意 }{k}_1{\boxed{\alpha }}_1+k_2{\boxed{\alpha }}_2+k_3{\boxed{\alpha }}_3=\boxed{\beta }\text{ 可以转换为线性方程组 }({\boxed{\alpha }}_1,{\boxed{\alpha }}_2,{\boxed{\alpha }}_3)\boxed{x}=\boxed{\beta }\text{，其中解}\\ & \text{向量 }\boxed{x}=\left(\begin{array}{c}{k}_1\\ k_2\\ k_3\end{array}\right)\text{，即}{\boxed{\alpha }}_1,{\boxed{\alpha }}_2,{\boxed{\alpha }}_3\text{ 线性表出}\boxed{\beta }\text{ 的系数；}\\ & ②{\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n\text{ 不能线性表出}\boxed{\beta }⟺({\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n)\boxed{x}=\boxed{\beta }\text{ 无解；}\\ & {\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n\text{ 能线性表出}\boxed{\beta }\text{，且表出方式唯一}⟺({\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n)\boxed{x}=\boxed{\beta }\text{ 有唯一解；}\\ & {\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n\text{ 能线性表出}\boxed{\beta }\text{，且表出方式不唯一}⟺({\boxed{\alpha }}_1,\cdots ,{\boxed{\alpha }}_n)\boxed{x}=\boxed{\beta }\text{ 有无穷多解。}\end{array}$$

选C