求解最大无关变量问题，线性代数 线性代数极大无关组
\u600e\u6837\u6c42\u6781\u5927\u65e0\u5173\u7ec4\uff0c\u7ebf\u6027\u4ee3\u6570\u95ee\u9898\uff0c\u5728\u7ebf\u6c42\u6559\uff01\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\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\u3002\u4f8b\u5b50\u5982\u4e0b\uff1a
\u6c42a1=(-1,-1,0,0)T\uff0ca2=(1,2,1,-2)T\uff0ca3=(0,1,1,-1)T\uff0ca4=(1,3,2,1)T\uff0ca5=(2,6,4,-1)T\u7684\u4e00\u4e2a\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u3002
-1 1 0 1 2
-1 2 1 3 6
0 1 1 2 4
0-1 -1 1 -1
\u5316\u7b80\u5f97:
A=
10 1 0 1
01 1 0 2
00 0 1 1
00 0 0 0
\u663e\u7136r(A)=3\u3002\u56e0\u6b64\u6781\u5927\u65e0\u5173\u7ec4\u67093\u4e2a\u5411\u91cf\u3002\u663e\u7136\u7b2c1,2,4\u5217\u4e3a\u5355\u4f4d\u77e9\u9635\u90e8\u5206,\u5bf9\u5e94\u7684\u5411\u91cf\u4e3aa1\uff0ca2\uff0ca4\u3002
\u6269\u5c55\u8d44\u6599
(1)\u53ea\u542b\u96f6\u5411\u91cf\u7684\u5411\u91cf\u7ec4\u6ca1\u6709\u6781\u5927\u65e0\u5173\u7ec4\u3002
(2)\u4e00\u4e2a\u7ebf\u6027\u65e0\u5173\u5411\u91cf\u7ec4\u7684\u6781\u5927\u65e0\u5173\u7ec4\u5c31\u662f\u5176\u672c\u8eab\u3002
(3)\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u5bf9\u4e8e\u6bcf\u4e2a\u5411\u91cf\u7ec4\u6765\u8bf4\u5e76\u4e0d\u552f\u4e00\u3002\u4f46\u662f\u6bcf\u4e2a\u5411\u91cf\u7ec4\u7684\u6781\u5927\u7ebf\u6027\u65e0\u5173\u7ec4\u90fd\u542b\u6709\u76f8\u540c\u4e2a\u6570\u7684\u5411\u91cf\u3002
(4)\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u89e3\u5411\u91cf\u7684\u6781\u5927\u65e0\u5173\u7ec4\u4e3a\u57fa\u7840\u89e3\u7cfb\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u6781\u5927\u65e0\u5173\u7ec4
\u4f60\u505a\u7684\u5b8c\u5168\u6b63\u786e\uff0c\u4ee3\u56de\u53bb\u9a8c\u8bc1\u4e5f\u662f\u6b63\u786e\u7684\u3002\u4e66\u4e0a\u7684\u7b54\u6848\u4ee3\u56de\u53bb\u4e0d\u5bf9\uff0c\u80af\u5b9a\u662f\u9519\u4e86\u3002\u987a\u4fbf\u8bf4\u4e00\u4e0b\uff0c\u8fd9\u7c7b\u95ee\u7b54\u7b54\u6848\u5e76\u4e0d\u552f\u4e00\uff0c\u4f8b\u5982\u4e5f\u53ef\u4ee5\u628a\u03b11\uff0c\u03b13\u4f5c\u4e3a\u6781\u5927\u65e0\u5173\u7ec4\u3002
A=(a1, a2, a3, a4, a5)[1 1 2 2 1]
[0 2 1 5 -1]
[2 0 3 -1 3]
[1 1 0 4 -1]
行初等变换为
[1 1 2 2 1]
[0 2 1 5 -1]
[0 -2 -1 -5 1]
[0 0 -2 2 -2]
行初等变换为
[1 1 2 2 1]
[0 2 1 5 -1]
[0 0 0 0 0]
[0 0 -2 2 -2]
行初等变换为
[1 1 0 4 -1]
[0 2 0 6 -2]
[0 0 1 -1 1]
[0 0 0 0 0]
行初等变换为
[1 0 0 1 0]
[0 1 0 3 -1]
[0 0 1 -1 1]
[0 0 0 0 0]
a1, a2, a3 为一个极大线性无关组
a4=a1+3a2-a3
a5=-a2+a3
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