2.A 张成空间与线性无关

本是古典何须时尚 2022-06-10 01:06 220阅读 0赞

### 向量组 ###

一组向量，通常不用括号，如![这里写图片描述][gif.latex_v_1_v_2_..._v_n]

### 线性组合 ###

![V][] 中的一组向量![向量组][gif.latex_v_1_v_2_..._v_n] 的线性组合为 ![线性组合][gif.latex_5Csmall_space_a_1v_1_a_2v_2_..._a_mv_m] 的向量，其中 ![条件][gif.latex_5Csmall_space_a_1_a_2_..._a_m_space_5Cin_space_F]

### 张成空间 ###

![V][]中的一组向量![向量组][gif.latex_v_1_v_2_..._v_m] 的**所有**线性组合构成的集合，称为 ![向量组][gif.latex_v_1_v_2_..._v_m] 的张成空间，记为 ![span][]  
即：  
![这里写图片描述][gif.latex_5Cinline_space_5Csmall_space_5Cbegin_7Baligned_7D_space_span_28v_1_..._v_m_29_5Cleft_space_5C_7B_space_a_1v_1_..._a_mv_m_a_1_..._a_m_5Cin_space_F_space_5Cright_space_5C_7D_5C_5C_space_span_28_28_29_29_5Cleft_space_5C_7B0_5Cright_space_5C_7D_space_5Cend_7Baligned_7D]

**张成空间是包含这组向量的最小子空间**  
这是简写，全称为：![V][] 中的一组向量的张成空间是包含这组向量的最小子空间。

### 张成 ###

如果 ![span][] 等于 ![V][]，则称为向量组张成 ![V][]比如  
![基][gif.latex_5Cinline_space_5Csmall_space_281_0_..._0_29_280_1_..._0_29_..._280_0_..._1_29] 张成 ![Fn][]

### 多项式 ###

多项式是一种映射：![映射][gif.latex_5Cinline_space_5Csmall_space_p_F_5Crightarrow_space_F] 如果  
![这里写图片描述][gif.latex_5Cinline_space_5Csmall_space_5Cbegin_7Baligned_7D_space_5Cexists_space_5C_space_a_0_a_1_space_..._a_m_space_5Cin_space_F_space_5C_5C_space_used_space_5C_space_5Cforall_space_z_space_5Cin_space_F_5C_5C_space_p_28z_29_a_0_a_1z_a_2z_5E2_..._a_mz_5Em_space_5Cend_7Baligned_7D]  
则称为 ![p][] 系数属于 ![F][] 的多项式

![PF][]是系数 ![F][] 的多项式集合

### 多项式的次数 ###

对于多项式 ![p][p 1] 若存在![标量][gif.latex_5Cinline_space_5Csmall_space_a_0_a_1_.._a_m_space_5Cin_space_F]，其中 ![这里写图片描述][gif.latex_5Cinline_space_5Csmall_space_a_m_space_5Cneq_space_0]，使得对于 ![多项式][gif.latex_5Cinline_space_5Csmall_space_5Cforall_space_z_space_5Cin_space_F_p_28z_29_a_0_a_1z_..._a_mz_5Em]  
则说的![p][p 2] **次数**为 ![V][V 1], 记 ![deg][]

![gif.latex_5Csmall_space_P_m_28F_29_0_5Cleq_space_m][] 表示系数![F][F 1]次数不超过![m][V 1]的所有多项式集合。它是有限维度的向量空间。

### 无限维度向量空间 ###

一个向量空间如果不是有限维度的，则是无限维度的。

### 线性无关 ###

![V][] 中一组向量 ![向量组][gif.latex_v_1_v_2_..._v_m] 称为线性无关，如果使得  
![线性无关][gif.latex_5Csmall_space_a_1v_1_..._a_mv_m_0_5C_space_a_1_..._a_m_5Cin_space_F_space_5C_space_and_space_5C_space_a_1_..._a_m_0]  
（1）一个向量线性无关，当且仅当 ![这里写图片描述][gif.latex_5Csmall_space_v_space_5Cneq_space_0]  
（2）两个向量线性无关，当且仅当每个向量不等于另外一个向量的倍数  
（3）![gif.latex_5Csmall_space_1_z_z_5E2_..._z_5Em_m_5Cgeqslant_space_0][] 线性无关

### 线性相关 ###

一组向量线性无关，则线性相关。即 ![向量组][gif.latex_5Csmall_space_v_1_v_2_..._v_m] 线性无关，当且仅当存在不全为0的 ![系数][gif.latex_5Csmall_space_a_1_a_2_..._a_m_space_5Cin_space_F] 使得 ![方程][gif.latex_5Csmall_space_a_1v_1_a_2v_m_..._a_mv_m_space_0]  
（1）一组向量中的某个向量是其余向量的线性组合，则这个向量组是线性相关的。因此，可知。一组线性无关的向量组中任意向量都不可能由**其余**向量线性表示。  
（2）包含0向量的向量组线性相关。因为0可以由其余向量线性表示。  
证明：如果一个向量是其余向量的线性组合，那么使得 ![方程][gif.latex_5Csmall_space_a_1v_1_a_2v_m_..._a_mv_m_space_0] 成立就无需所有系数为0，证毕。  
**引理**  
![向量组][gif.latex_5Csmall_space_v_1_v_2_..._v_m] 是 ![V][V 2] 线性无关的向量组，则有 ![这里写图片描述][gif.latex_5Csmall_space_j_5Cin_space_5C_7B1_2_..._m_5C_7D] 使得  
（a）![这里写图片描述][gif.latex_5Csmall_space_v_j_5Cin_space_5Crm_space_span_28_5Ctextit_space_v_1_..._5Ctextit_space_v_7Bj-1_7D_29]  
（b）![向量组][gif.latex_5Csmall_space_v_1_v_2_..._v_m] 中去掉 *j* 项，则剩余组成的张成空间等于 ![span][span 1]  
换言之，存在一个向量可以有其余向量线性表示并且属于其余向量的张成空间。  
**线性无关组的长度小于等于张成组的长度**

### 有限维度子空间 ###

有限维度向量空间的子空间都是有限维度的。

附录：  
本节重要的公式latex文本和url  
p\\in P(F)  
a\_0,a\_1,..,a\_m \\in F  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\inline&space;\\small&space;a\_0,a\_1,..,a\_m&space;\\in&space;F

a\_m \\neq 0  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\inline&space;\\small&space;a\_m&space;\\neq&space;0

\\forall z \\in F;p(z)=a\_0+a\_1z+…+a\_mz^m  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\inline&space;\\small&space;\\forall&space;z&space;\\in&space;F;p(z)=a\_0+a\_1z+…+a\_mz^m

\\rm deg p=m  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\inline&space;\\small&space;\\rm&space;deg&space;p=m

P\_m(F):0\\leq m  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\small&space;P\_m(F):0\\leq&space;m

a\_1v\_1+…+a\_mv\_m=0;\\ a\_1,…,a\_m\\in F \\ and \\ a\_1=…=a\_m=0  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\small&space;a\_1v\_1+…+a\_mv\_m=0;\\&space;a\_1,…,a\_m\\in&space;F&space;\\&space;and&space;\\&space;a\_1=…=a\_m=0

v\_j\\in \\rm span(\\textit v\_1,…,\\textit v\_\{j-1\})  
[http://latex.codecogs.com/gif.latex][http_latex.codecogs.com_gif.latex]?\\small&space;\\rm&space;span(\\textit&space;v\_1,…,\\textit&space;v\_\{j-1\})

[gif.latex_v_1_v_2_..._v_n]: /images/20220610/1c49ecdb487c4e449dd667d38aa0ada0.png
[V]: /images/20220610/567ede9b0a6547fd840ec7ca1d28db31.png
[gif.latex_5Csmall_space_a_1v_1_a_2v_2_..._a_mv_m]: http://latex.codecogs.com/gif.latex?%5Csmall&space;a_1v_1+a_2v_2+...+a_mv_m
[gif.latex_5Csmall_space_a_1_a_2_..._a_m_space_5Cin_space_F]: http://latex.codecogs.com/gif.latex?%5Csmall&space;a_1,a_2,...,a_m&space;%5Cin&space;F
[gif.latex_v_1_v_2_..._v_m]: http://latex.codecogs.com/gif.latex?v_1,v_2,...,v_m
[span]: /images/20220610/b86093f9dfe24f99aecd7dcf2288b36e.png
[gif.latex_5Cinline_space_5Csmall_space_5Cbegin_7Baligned_7D_space_span_28v_1_..._v_m_29_5Cleft_space_5C_7B_space_a_1v_1_..._a_mv_m_a_1_..._a_m_5Cin_space_F_space_5Cright_space_5C_7D_5C_5C_space_span_28_28_29_29_5Cleft_space_5C_7B0_5Cright_space_5C_7D_space_5Cend_7Baligned_7D]: /images/20220610/c5980918e1fd447bb96dcf66ddbc9687.png
[gif.latex_5Cinline_space_5Csmall_space_281_0_..._0_29_280_1_..._0_29_..._280_0_..._1_29]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;%281,0,...,0%29,%280,1,...,0%29,...,%280,0,...,1%29
[Fn]: /images/20220610/84703c6a3f9b40be84a5ead2da6a8b88.png%5En:%28x_1,x_2,...,x_n%29=x_1%281,0,...,0%29+x_2%280,1,...,0%29+...+x_n%280,0,...,1%29
[gif.latex_5Cinline_space_5Csmall_space_p_F_5Crightarrow_space_F]: /images/20220610/952f612fadfa4acea9d63737cefd2b36.png
[gif.latex_5Cinline_space_5Csmall_space_5Cbegin_7Baligned_7D_space_5Cexists_space_5C_space_a_0_a_1_space_..._a_m_space_5Cin_space_F_space_5C_5C_space_used_space_5C_space_5Cforall_space_z_space_5Cin_space_F_5C_5C_space_p_28z_29_a_0_a_1z_a_2z_5E2_..._a_mz_5Em_space_5Cend_7Baligned_7D]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;%5Cbegin%7Baligned%7D&space;%5Cexists&space;%5C&space;a_0,a_1,&space;...,a_m&space;&%5Cin&space;F&space;%5C%5C&space;used&space;%5C&space;%5Cforall&space;z&space;&%5Cin&space;F%5C%5C&space;p%28z%29&=a_0+a_1z+a_2z%5E2+,...,+a_mz%5Em&space;%5Cend%7Baligned%7D
[p]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;p
[F]: /images/20220610/84703c6a3f9b40be84a5ead2da6a8b88.png
[PF]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;P%28F%29
[p 1]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;p%5Cin&space;P%28F%29
[gif.latex_5Cinline_space_5Csmall_space_a_0_a_1_.._a_m_space_5Cin_space_F]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;a_0,a_1,..,a_m&space;%5Cin&space;F
[gif.latex_5Cinline_space_5Csmall_space_a_m_space_5Cneq_space_0]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;a_m&space;%5Cneq&space;0
[gif.latex_5Cinline_space_5Csmall_space_5Cforall_space_z_space_5Cin_space_F_p_28z_29_a_0_a_1z_..._a_mz_5Em]: http://latex.codecogs.com/gif.latex?%5Cinline&space;%5Csmall&space;%5Cforall&space;z&space;%5Cin&space;F;p%28z%29=a_0+a_1z+...+a_mz%5Em
[p 2]: http://latex.codecogs.com/gif.latex?p
[V 1]: http://latex.codecogs.com/gif.latex?m
[deg]: /images/20220610/dc65655872fd484e95f6d9df5ba866ef.png
[gif.latex_5Csmall_space_P_m_28F_29_0_5Cleq_space_m]: http://latex.codecogs.com/gif.latex?%5Csmall&space;P_m%28F%29:0%5Cleq&space;m
[F 1]: http://latex.codecogs.com/gif.latex?F
[gif.latex_5Csmall_space_a_1v_1_..._a_mv_m_0_5C_space_a_1_..._a_m_5Cin_space_F_space_5C_space_and_space_5C_space_a_1_..._a_m_0]: /images/20220610/972face2a60949bf974ad6374fe25cef.png
[gif.latex_5Csmall_space_v_space_5Cneq_space_0]: /images/20220610/f17460ca073f4144803d01d52f366517.png
[gif.latex_5Csmall_space_1_z_z_5E2_..._z_5Em_m_5Cgeqslant_space_0]: /images/20220610/5d1e4fc5249146b493fc6e9b2a96690f.png
[gif.latex_5Csmall_space_v_1_v_2_..._v_m]: http://latex.codecogs.com/gif.latex?%5Csmall&space;v_1,v_2,...,v_m
[gif.latex_5Csmall_space_a_1v_1_a_2v_m_..._a_mv_m_space_0]: http://latex.codecogs.com/gif.latex?%5Csmall&space;a_1v_1+a_2v_m+...+a_mv_m&space;=0
[V 2]: http://latex.codecogs.com/gif.latex?%5Csmall&space;V
[gif.latex_5Csmall_space_j_5Cin_space_5C_7B1_2_..._m_5C_7D]: /images/20220610/2b5ad1f4dd174073954087d4759ee007.png
[gif.latex_5Csmall_space_v_j_5Cin_space_5Crm_space_span_28_5Ctextit_space_v_1_..._5Ctextit_space_v_7Bj-1_7D_29]: /images/20220610/d2f9d24982be4dde908dcdd6b775e002.png
[span 1]: /images/20220610/9a29d9656d7c426fa8b9d0ab8ab442eb.png
[http_latex.codecogs.com_gif.latex]: http://latex.codecogs.com/gif.latex