线性代数中极大无关组的问题 线性代数 极大线性无关组的问题
\u7ebf\u6027\u4ee3\u6570 \u6781\u5927\u65e0\u5173\u7ec4\u7684\u95ee\u9898\u53ea\u8981\u662f\u8fdb\u884c\u521d\u7b49\u53d8\u6362\u5316\u4e3a\u9636\u68af\u578b\uff0c\u5c31\u53ef\u4ee5\u770b\u51fa\u77e9\u9635\u7684\u79e9\uff0c\u7136\u540e\u7531\u4e09\u79e9\u76f8\u7b49\uff0c\u53ef\u4ee5\u770b\u51fa\u884c\u5411\u91cf\u6216\u5217\u5411\u91cf\u7ec4\u4e2d\u7684\u7ebf\u6027\u65e0\u5173\u7684\u5411\u91cf\uff0c\u5e76\u4e14\u5176\u4f59\u5411\u91cf\u53ef\u7531\u4ed6\u4eec\u8868\u793a\u662f\u663e\u7136\u7684\u3002
\u671b\u91c7\u7eb3
\u5475\u5475\uff0c\u5f88\u7b80\u5355\u554a\u3002
\u5148\u628a\u90a3\u51e0\u4e2a\u5411\u91cf\u4ee5\u5217\u5411\u91cf\u7684\u5f62\u5f0f\u5199\u6210\u4e00\u4e2a\u77e9\u9635\uff0c\u7136\u540e\u6c42\u8fd9\u4e2a\u77e9\u9635\u7684\u79e9\uff0c\u56e0\u4e3a\u6781\u5927\u65e0\u5173\u7ec4\u4e2d\u5411\u91cf\u7684\u4e2a\u6570\u5c31\u662f\u77e9\u9635\u7684\u79e9\u3002\u8981\u6c42\u77e9\u9635\u7684\u79e9\u5f53\u7136\u8981\u5148\u628a\u77e9\u9635\u5316\u6210\u884c\u7b80\u5316\u9636\u68af\u578b\u77e9\u9635\u5566\uff0c\u7136\u540e\u770b\u770b\u5176\u4e2d\u7684\u5355\u4f4d\u9635\u90e8\u5206\u5bf9\u5e94\u54ea\u51e0\u4e2a\u5411\u91cf\uff0c\u8fd9\u51e0\u4e2a\u5411\u91cf\u4fbf\u662f\u6781\u5927\u65e0\u5173\u7ec4\u7684\u6210\u5458\u55bd~\u3002\u4f8b\u5b50\u5982\u4e0b\uff1a
\u6c42a1=(-1,-1,0,0)T a2=(1,2,1,-2)T a3=(0,1,1,-1)T a4=(1,3,2,1)T
a5=(2,6,4,-1)T \u7684\u4e00\u4e2a\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u3002
\u89e3\uff1aA=
-1 1 0 1 2
-1 2 1 3 6
0 1 1 2 4
0 -1 -1 1 -1
\u5316\u7b80\u5f97\uff1a
A=
1 0 1 0 1
0 1 1 0 2
0 0 0 1 1
0 0 0 0 0
\u663e\u7136r\uff08A)=3.\u56e0\u6b64\u6781\u5927\u65e0\u5173\u7ec4\u67093\u4e2a\u5411\u91cf\u3002
\u663e\u7136\u7b2c1\uff0c2\uff0c4\u5217\u4e3a\u5355\u4f4d\u77e9\u9635\u90e8\u5206\uff0c\u5bf9\u5e94\u7684\u5411\u91cf\u4e3aa1 a2 a4,
\u56e0\u6b64\u6b64\u5373\u4e3a\u6781\u5927\u65e0\u5173\u7ec4\u3002
0 3 1 2 0 1 3 2 0 1 3 2 0 3 2 —> 0 3 2 1 3 2 —>0 7/2 0
0 0 0 0 —>0 0 0 0 —> 0 0 -1 -4 0 -1 -4 0 0 -10/3 0 -1 -4 0 -1 -4
0 -1 0 -4 0 0 -1 -4 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
所以此矩阵的秩为3，极大无关组是α1,α2,α3;或α1,α3,α4;或α1,α2,α4;或α2,α3,α4.
a1,a2,a3 是一个极大无关组.
注: 想像一下将2,3列交换,就是一个梯矩阵而不必再化简
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