写出原方程组的通解对含参数线性方程组解的讨论一方程个数=未知量个数1行列式法2初等变换法二方程个数不
xie chu yuan fang cheng zu de tong jie dui han can shu xian xing fang cheng zu jie de tao lun yi fang cheng ge shu = wei zhi liang ge shu 1 hang lie shi fa 2 chu deng bian huan fa er fang cheng ge shu bu
增广矩阵 化阶梯形2由阶梯形求特解及对应齐次方程组基础解系（线性无关的）3按非齐次线性方程组解的结构
zeng guang ju zhen hua jie ti xing 2 you jie ti xing qiu te jie ji dui ying qi ci fang cheng zu ji chu jie xi xian xing wu guan de 3 an fei qi ci xian xing fang cheng zu jie de jie gou
穷多解解线性方程组常用以下3个方法1克莱姆法则2消元法3行等价标准形求非齐次线性方程组通解方法1写出
qiong duo jie jie xian xing fang cheng zu chang yong yi xia 3 ge fang fa 1 ke lai mu fa ze 2 xiao yuan fa 3 hang deng jia biao zhun xing qiu fei qi ci xian xing fang cheng zu tong jie fang fa 1 xie chu
关解向量非齐次r(a)<r(b)无解r(a)=r(b)=n百度快照唯一解r(a)=r(b)<n无
guan jie xiang liang fei qi ci r(a)<r(b) wu jie r(a)=r(b)=n bai du kuai zhao wei yi jie r(a)=r(b)<n wu
解当r(a)=r<n无穷多解有r个独立未知量、r个独立方程、n-r个自由未知量、n-r个线性无
jie dang r(a)=r<n wu qiong duo jie you r ge du li wei zhi liang r ge du li fang cheng n-r ge zi you wei zhi liang n-r ge xian xing wu
lt;=n证明a=0常用方法 a百度快照ij=0r(a)=0四线性方程组解的结构齐次当r(a)=r=n唯一0
lt;=n zheng ming a=0 chang yong fang fa a bai du kuai zhao ij=0r(a)=0 si xian xing fang cheng zu jie de jie gou qi ci dang r(a)=r=n wei yi 0
(r(a),r(b))r(ab)>=r(a)+r(b)-n特别的ab=0 r(a)+r(b)&
(r(a),r(b))r(ab)>=r(a)+r(b)-n te bie de ab=0 r(a)+r(b)&
换变成矩阵b线性关系相同注：对以列向量按列构成的矩阵做初等行变换三矩阵的秩r(ab)<=min
huan bian cheng ju zhen b xian xing guan xi xiang tong zhu dui yi lie xiang liang an lie gou cheng de ju zhen zuo chu deng hang bian huan san ju zhen de zhi r(ab)<=min
性相关性求向量组极大百度快照线性无关组1逐个选录法2消元法3利用矩阵初等变换法矩阵a通过有限次初等行（列）变
xing xiang guan xing qiu xiang liang zu ji da bai du kuai zhao xian xing wu guan zu 1 zhu ge xuan lu fa 2 xiao yuan fa 3 li yong ju zhen chu deng bian huan fa ju zhen a tong guo you xian ci chu deng hang lie bian
，任何部分线性无关向量组秩=行秩=列秩若是方阵，可通过计算行列式来判断利用线性方程组有无0解来判断线
ren he bu fen xian xing wu guan xiang liang zu zhi = hang zhi = lie zhi ruo shi fang zhen ke tong guo ji suan hang lie shi lai pan duan li yong xian xing fang cheng zu you wu 0 jie lai pan duan xian
量成比例3多于n百度快照个的n维向量必定线性相关4一部分向量线性相关，整个向量线性相关；若整个向量组线性无关
liang cheng bi li 3 duo yu n bai du kuai zhao ge de n wei xiang liang bi ding xian xing xiang guan 4 yi bu fen xiang liang xian xing xiang guan zheng ge xiang liang xian xing xiang guan ruo zheng ge xiang liang zu xian xing wu guan
等价的向量组秩相等1单独一个0向量线性相关，而一个非0向量线性无关2两个向量线性相关充要条件是对应分
deng jia de xiang liang zu zhi xiang deng 1 dan du yi ge 0 xiang liang xian xing xiang guan er yi ge fei 0 xiang liang xian xing wu guan 2 liang ge xiang liang xian xing xiang guan chong yao tiao jian shi dui ying fen
解d=0再对增广矩阵做初等行变换判断如果纯用初等变换百度快照法，可能产生讨论不全的错误二 n维向量线性相关性
jie d=0 zai dui zeng guang ju zhen zuo chu deng hang bian huan pan duan ru guo chun yong chu deng bian huan bai du kuai zhao fa ke neng chan sheng tao lun bu quan de cuo wu er n wei xiang liang xian xing xiang guan xing
非0解、无穷多解在讨论带参数线性方程组时；若方程个数与未知量个数相等最好先用克莱姆法则d不=0有唯一
fei 0 jie wu qiong duo jie zai tao lun dai can shu xian xing fang cheng zu shi ruo fang cheng ge shu yu wei zhi liang ge shu xiang deng zui hao xian yong ke lai mu fa ze d bu =0 you wei yi
出组=方程组对应的齐次方程组百度快照变换后方程与原方程同解方程组的导出组必定有解r=n唯一0解r<n有
chu zu = fang cheng zu dui ying de qi ci fang cheng zu bai du kuai zhao bian huan hou fang cheng yu yuan fang cheng tong jie fang cheng zu de dao chu zu bi ding you jie r=n wei yi 0 jie r<n you
一 消元法线性方程组的初等变换1用一非0数乘某一方程2把一方程倍数加到另一方程3互换2方程位置注：导
yi xiao yuan fa xian xing fang cheng zu de chu deng bian huan 1 yong yi fei 0 shu cheng mou yi fang cheng 2 ba yi fang cheng bei shu jia dao ling yi fang cheng 3 hu huan 2 fang cheng wei zhi zhu dao