数学实验10.6线性代数实验(二).docx

10.6
%第一大题
%1.1
m=4;
A=[1 -2 -1 2;-2 4 2 -6;2 -1 0 3;3 3 3 4];
AV=rank(A);
if m==AV
Z='向量组A线性无关'
else Z='向量组A线性相关'
end
A1=rref(A)
%Z =

向量组A线性相关

A1 =

1.0000 0 0.3333 0
0 1.0000 0.6667 0
0 0 0 1.0000
0 0 0 0
%向量组的秩为2；向量组A为线性相关向量组；a1和a2为一个极大无关组；
a3=
%1.2
B=fix(30*rand(6));
for i=1:6
if B(i,i)~=0
for j=i+1:6
B(j,:)=-B(i,:)*B(j,i)/B(i,i)+B(j,:);
end
else
for k=i+1:6
if B(k,i)~=0
C=B(i,:);
B(i,:)=B(k,:);
B(k,:)=C;
break;
end
end
end
end
B
%B =

19.0000 20.0000 10.0000 27.0000 9.0000 14.0000
0 -27.4737 8.2632 -39.7895 9.7368 -16.6316
0 0 16.2050 3.5862 10.5536 5.0651
0 0 0 8.7107 -4.0046 17.0807
0 0 0 0 -14.6511 23.8761
0 0 0 0 0 37.5963
%1.3
A=[1 2 4 6 -3 2;2 4 -4 5 1 -5;3 6 2 0 5 -9;2 3 0 4 0 1;0 -4 -5 2 1 4;5 5 -3 6 6 -4];
b=[2 ;-1; 3; 4 ;-2; 5];
B=[A b];
R1=rank(A)
R2=rank(B)
if R1~=R2
Z='方程组无解'
else
Z='方程组有解'
end
RR=rref(B)
%R1 =5 R2 =5
Z =方程组有解
RR =

1.0000 0 0 0 0 6.8068 6.6809
0 1.0000 0 0 0 -2.2520 -0.9342
0 0 1.0000 0 0 -0.2041 0.0315
0 0 0 1.0000 0 -1.4644 -1.6397
0 0 0 0 1.0000 -3.1000 -2.3000
0 0 0 0 0 0 0
%1.4
A8=magic(5)
[D,X]=eig(A8)
%A8 =

17 24 1 8 15
23 5 7 14 16
4 6 13 20 22
10 12 19 21 3
11 18 25 2 9

D =

-0.4472 0.0976 -0.6330 0.6780 -0.2619
-0.4472 0.3525 0.5895 0.3223 -0.1732
-0.4472 0.5501 -0.3915 -05501 0.3915
-0.4472 -0.3223 0.1732 -0.3525 -0.5895
-0.4472 -0.6780 0.2619 -0.0976 0.6330

X =

65.0000 0 0 0 0
0 -21.2768 0 0 0
0 0 -13.1263 0 0
0 0 0 21.2768 0
0 0 0 0 13.1263
%1.5
A=[1 1 -2 -3 0 0 ;1 2 1 4 0 3;-2 1 -1 2 -3 1;-3 4 2 1 -1 2;0 0 -3 -1 -4 4;0 3 1 2 4 -1];
[D,X]=eig(A)
P=orth(D)
%D =

-0.1125 -0.4801 -0.1592 0.5581 -0.5933 0.2609
-0.0548 0.5706 0.1797 0.3749 -0.4021 -0.5805
-0.3665 0.0003 -0.7518 0.2573 0.3826 -0.2966
-0.1251 -0.6457 0.3949 0.0057 0.1740 -0.6175
-0.7539 0.0259 -0.0112 -0.5372 -0.3771 0.0073
0.5158 -0.1627 -0.4703 -0.4394 -0.4093 -0.3545

X =

-8.3609 0 0 0 0 0
0 -4.2224 0 0 0 0
0 0 -2.1320 0 0 0
0 0 0 0.7192 0 0
0 0 0 0 3.8471 0
0 0 0 0 0 8.1489

P =

0.1296 0.7932 0.0522 -0.1125 -0.0015 0.5819
-0.0065 0.2540 0.8375 -0.0548 -0.2123 -0.4311
0.5965 0.2509 -0.3997 -0.3665 -0.1639 -0.5103
0.6024 -0.4672 0.2838 -0.1251 -0.3436 0.4523
-0.4976 -0.0946 -0.0744 -0.7539 -0.3994 0.0998
-0.1299 0.1267 -0.2235 0.5158 -0.8065 -0.0260
其标准型为：f=-8.3609*y1^2 -4.2224*y2^2-2.1320*y3^2+0.7192*y4^2+3.8471*y5^2+8.1489*y6^2
%第二大题
(1)A=[3 1;0 5];
B=[3 9 25 8 0 11 20 7 10 8 7;0 8 20 5 13 8 18 8 0 18 8];
C=A*B;
C1=rem(C,26)
%C1 =

9 9 17 3 13 15 0 3 4 16 3
0 14 22 25 13 14 12 14 0 12 14
%明文为：jajorwdznnpoamdoeaqmdo
(2)ad=det(A);
a1=[5 -1;0 3];
a2=11*a1+ad*26;
a3=rem(a2,26);
c1=[25 3 16 11 4 11 1 8;14 14 16 13 8 13 5 14];
c2=a3*c1;
b2=rem(c2,26)
%b2 =
25 11 2 20 2 20 0 0
20 20 8 13 4 13 9 20
%密文为zuluciunceunajau