如何用Matlab求线性方程组的通解 如何用matlab求解齐次线性方程组
\u5982\u4f55\u7528Matlab\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u901a\u89e3\u7ed9\u4f60\u4e00\u4e2a\u4f8b\u5b50\uff0c\u6765\u8bf4\u660e\u5982\u4f55\u7528Matlab\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u901a\u89e3\u3002
>> a=[1 -1 1 -1;-1 1 1 -1;2 -2 -1 1]; %\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635
>> b=[1;1;-1]; % \u5e38\u5217\u5411\u91cf
>> [rank(a) rank([a,b])]
ans =
2 2 %\u79e9\u76f8\u7b49\u4e14\u5c0f\u4e8e4\uff0c\u8bf4\u660e\u6709\u65e0\u7a77\u591a\u89e3
>> rref([a,b]) %\u7b80\u5316\u884c\u9636\u68af\u5f62\u77e9\u9635
ans =
1 -1 0 0 0
0 0 1 -1 1
0 0 0 0 0
\u4ece\u800c\u539f\u65b9\u7a0b\u7ec4\u7b49\u4ef7\u4e8ex1=x2\uff0cx3=x4+1\u3002
\u4ee4x2=k1\uff0cx4=k2
\u4e8e\u662f\uff0c\u6211\u4eec\u6c42\u5f97\u901a\u89e3
\u5148\u5199m\u6587\u4ef6
function [x,y]=line_solution(A,b)
[m,n]=size(A);
y=[];
if norm(b)>0
if rank(A)==rank([A,b])
if rank(A)==n
disp('\u65b9\u7a0b\u6709\u552f\u4e00\u89e3x');
x=A\b;
else
disp('\u65b9\u7a0b\u6709\u65e0\u7a77\u591a\u89e3,\u7279\u89e3\u4e3ax\uff0c\u5176\u9f50\u6b21\u65b9\u7a0b\u7ec4\u7684\u57fa\u7840\u89e3\u7cfb\u4e3ay');
x=A\b;
y=null(A,'r');%null\u662f\u7528\u6765\u6c42\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u57fa\u7840\u89e3\u7cfb\u7684\uff0c\u52a0\u4e0a'r'\u5219\u6c42\u51fa\u7684\u662f\u4e00\u7ec4\u6700\u5c0f\u6b63\u6574\u6570\u89e3\uff0c\u5982\u679c\u4e0d\u52a0\uff0c\u5219\u6c42\u51fa\u7684\u662f\u89e3\u7a7a\u95f4\u7684\u89c4\u8303\u6b63\u4ea4\u57fa\u3002
end
else
disp('\u65b9\u7a0b\u65e0\u89e3');
x=[];
end
else
disp('\u539f\u65b9\u7a0b\u7ec4\u6709\u552f\u4e00\u96f6\u89e3x');
x=zeros(n,1);
if rank(A)<n
disp('\u65b9\u7a0b\u7ec4\u6709\u65e0\u7a77\u4e2a\u89e3\uff0c\u57fa\u7840\u89e3\u7cfb\u4e3ay');
y=null(A,'r');
end
end
----------------------------------------------------------------------
\u4e3e\u4f8b\u8c03\u7528\uff1a
format rat %\u4ee5\u6709\u7406\u6570\u5f62\u5f0f\u8f93\u51fa
A=[1,1,-3,-1;3,-1,-3,4;1,5,-9,-8];
b=[1;4;0];
[x,y]=line_solution(A,b);
x,y
format short %\u4fdd\u75594\u4f4d\u6709\u6548\u6570\u5b57
>> a=[1 -1 1 -1;-1 1 1 -1;2 -2 -1 1]; %线性方程组的系数矩阵
>> b=[1;1;-1]; % 常列向量
>> [rank(a) rank([a,b])]
ans =
2 2 %秩相等且小于4，说明有无穷多解
>> rref([a,b]) %简化行阶梯形矩阵
ans =
1 -1 0 0 0
0 0 1 -1 1
0 0 0 0 0
从而原方程组等价于x1=x2，x3=x4+1。
令x2=k1，x4=k2
于是，我们求得通解
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