求非齐次线性方程组全部解并用导出组的基础解系表示 x1+x2=5 2x1+x2+x3+2x4=1 求非齐次线性方程组x1+2x2-x3+3x4=3,2x1+5...
\u7ebf\u6027\u4ee3\u6570\u9898\uff0c\u6c42\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u901a\u89e3\u5e76\u7528\u5176\u5bfc\u51fa\u7ec4\u7684\u57fa\u7840\u89e3\u7cfb\u8868\u793a\uff0c\u8981\u8be6\u7ec6\u89e3\u7b54\u8fc7\u7a0b\uff0c\u6700\u540e\u53d1\u56fe\u7247\u6e05\u695a\u4e00\u70b9\u589e\u5e7f\u77e9\u9635 (A, b) =
[1 2 3 1 -3 5]
[2 1 0 2 -6 1]
[3 4 5 6 -3 12]
[1 1 1 3 1 4]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 2 3 1 -3 5]
[0 -3 -6 0 0 -9]
[0 -2 -4 3 6 -3]
[0 -1 -2 2 4 -1]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 0 -1 1 -3 -1]
[0 1 2 0 0 3]
[0 0 0 3 6 3]
[0 0 0 2 4 2]
\u884c\u521d\u7b49\u53d8\u6362\u4e3a
[1 0 -1 0 -5 -2]
[0 1 2 0 0 3]
[0 0 0 1 2 1]
[0 0 0 0 0 0]
r(A,b) = r(A) = 3<5, \u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u591a\u89e3\u3002
\u65b9\u7a0b\u7ec4\u540c\u89e3\u53d8\u5f62\u4e3a
x1 = -2+x3+5x5
x2 = 3-2x3
x4 = 1-2x5
\u53d6 x3=x5=0, \u5f97\u7279\u89e3 (-2 3 0 1 0)^T,
\u5bfc\u51fa\u7ec4\u4e3a
x1 = x3+5x5
x2 = -2x3
x4 = -2x5
\u53d6 x3=1\uff0cx5=0, \u5f97\u57fa\u7840\u89e3\u7cfb (1 -2 1 0 0)^T,
\u53d6 x3=0\uff0cx5=1, \u5f97\u57fa\u7840\u89e3\u7cfb (5 0 0 -2 1)^T,
\u5219\u65b9\u7a0b\u7ec4\u7684\u901a\u89e3\u662f
x = (-2 3 0 1 0)^T+ k (1 -2 1 0 0)^T
+ c (5 0 0 -2 1)^T,
\u5176\u4e2d k\uff0c c \u4e3a\u4efb\u610f\u5e38\u6570\u3002
\u5177\u4f53\u56de\u7b54\u89c1\u56fe\uff1a
\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662frank(A)=n\u3002
\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u591a\u89e3\u7684\u5145\u8981\u6761\u4ef6\u662frank(A)<n\u3002\uff08rank(A)\u8868\u793aA\u7684\u79e9\uff09
\u6269\u5c55\u8d44\u6599\uff1a
\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4Ax=b\u7684\u6c42\u89e3\u6b65\u9aa4\uff1a
\uff081\uff09\u5bf9\u589e\u5e7f\u77e9\u9635B\u65bd\u884c\u521d\u7b49\u884c\u53d8\u6362\u5316\u4e3a\u884c\u9636\u68af\u5f62\u3002\u82e5R(A)<R(B)\uff0c\u5219\u65b9\u7a0b\u7ec4\u65e0\u89e3\u3002
\uff082\uff09\u82e5R(A)=R(B)\uff0c\u5219\u8fdb\u4e00\u6b65\u5c06B\u5316\u4e3a\u884c\u6700\u7b80\u5f62\u3002
\uff083\uff09\u8bbeR(A)=R(B)=r\uff1b\u628a\u884c\u6700\u7b80\u5f62\u4e2dr\u4e2a\u975e\u96f6\u884c\u7684\u975e0\u9996\u5143\u6240\u5bf9\u5e94\u7684\u672a\u77e5\u6570\u7528\u5176\u4f59n-r\u4e2a\u672a\u77e5\u6570\uff08\u81ea\u7531\u672a\u77e5\u6570\uff09\u8868\u793a\u3002
\u5bf9\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u65bd\u884c\u521d\u7b49\u884c\u53d8\u6362\u5316\u4e3a\u9636\u68af\u578b\u77e9\u9635\u540e\uff0c\u4e0d\u5168\u4e3a\u96f6\u7684\u884c\u6570r\uff08\u5373\u77e9\u9635\u7684\u79e9\uff09\u5c0f\u4e8e\u7b49\u4e8em\uff08\u77e9\u9635\u7684\u884c\u6570\uff09\uff0c\u82e5mr,\u5219\u5176\u5bf9\u5e94\u7684\u9636\u68af\u578bn-r\u4e2a\u81ea\u7531\u53d8\u5143\uff0c\u8fd9\u4e2an-r\u4e2a\u81ea\u7531\u53d8\u5143\u53ef\u53d6\u4efb\u610f\u53d6\u503c\uff0c\u4ece\u800c\u539f\u65b9\u7a0b\u7ec4\u6709\u975e\u96f6\u89e3\uff08\u65e0\u7a77\u591a\u4e2a\u89e3\uff09\u3002
\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e24\u4e2a\u89e3\u7684\u548c\u4ecd\u662f\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e00\u7ec4\u89e3\u3002\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u7684k\u500d\u4ecd\u7136\u662f\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u3002\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u79e9r(A)=n\uff0c\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u96f6\u89e3\u3002\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u79e9r(A)<n,\u65b9\u7a0b\u7ec4\u6709\u65e0\u6570\u591a\u89e3\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1\u2014\u2014\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4
1 1 0 0 5
2 1 1 2 1
5 3 2 2 3
r2-2r1,r3-5r1
1 1 0 0 5
0 -1 1 2 -9
0 -2 2 2 -22
r1+r2,r2*(-1),r3+2r2
1 0 1 2 -4
0 1 -1 -2 9
0 0 0 -2 -4
r1+r3,r2-r3,r3*(-1/2)
1 0 1 0 -8
0 1 -1 0 13
0 0 0 1 2
非齐次线性方程组的一个解:(-8,13,0,2)^T
对应的齐次线性方程组的基础解系:(-1,1,1,0)^T
方程组的所有解为:(-8,13,0,2)^T + c(-1,1,1,0)^T
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