% Problem 1 format short % (a) A = [ 1 1 1 1; 1 cos(0.1) cos(0.1)^2 cos(0.1)^3; 1 cos(0.2) cos(0.2)^2 cos(0.2)^3; 1 cos(0.3) cos(0.3)^2 cos(0.3)^3 ] % (b) rref(A) det(A) % (c) % If rref(A) is the identity matrix, then there is no free variable, so Ax = 0 only has trivial solution % If det(A) is nonzero, then A is invertible % (d) B = [ 1 1 1 1; 1 cos(0.2) cos(0.2)^2 cos(0.2)^3; 1 cos(0.5) cos(0.5)^2 cos(0.5)^3; 1 cos(1) cos(1)^2 cos(1)^3 ] det(B) % (e) % 1 = sin(t)^2 + cos(t)^2 % (f) % The determinant will be zero clear; % Problem 2 format rat; % (a) A = [ 0 3 1 8 1; -9 -12 4 11 -2; -3 -5 1 1 -1; -6 -4 4 18 0 ] rank(A) % (b) % dim(Nul A) = 3, dim(Col A)= 2, dim(Row A) = 2 % (c) rref(A) % (i) A basis for Nul A is % {[8/9; -1/3; 1; 0; 0], [43/9; -8/3; 0; 1; 0], [2/9; -1/3; 0; 0; 1]} % (ii) A basis for Col A is % {[0; -9; -3; -6], [3; -12; -5; -4]} % (iii) A basis for Row A is % {[1 0 -8/9 -43/9 -2/9], [0 1 1/3 8/3 1/3]} clear; % Problem 3 format rat; % (a) v1 = [0; 2; -4; -3; 1; 0] v2 = [1; 2; -12; 3; -2; 2] v3 = [2; 1; -3; -2; 2; 4] v4 = [2; -2; 0; 5; -1; 4] % (b) A = [v1 v2 v3 v4] rref(A) % (c) % A4 = -9/5 * A1 + 2/5 * A2 + 4/5 * A3 clear; % Problem 4 format rat; % (a) P = [ 1 1 2 4; 0 2 -1 -1; 0 0 3 0; 0 0 0 1 ] Q = [ 1 2 0 0; 3 1 3 3; 0 0 -1 0; 1 0 4 0 ] % (b) R = Q^-1 * P % (c) Q1 = Q^-1 Q1(:,[4]) % (d) v = [1; 2; 3; 4] R * v % (e) syms t [1 t t^2 t^3] * P * v clear; % Problem 5 format rat; % (a) A = [ 7 -6 20; -18 15 -50; -8 7 -23 ] A^2 A^3 A^4 A^5 A^6 A^7 A^8 % (b) [P, D] = eig(A) % (c) D^2 D^3 D^4 % D^4 = I % (d) % A^20000001 = PD^20000001P^-1 = PDP^-1 = A clear; % Problem 6 format short; % (a) v1 = [-14; 13; 7; 0; 3] v2 = [-7; -1; 14; 2; 1] v3 = [-8; 0; 4; 6; -3] v4 = [-3; 15; -13; 12; 13] % (b) A = [v1 v2 v3 v4] rank(A) % Because the dimension of the column space is the same as the rank, which shows that these four vectors are linearly independent % (c) w1 = v1 w2 = v2 - (dot(w1,v2)/dot(w1,w1)) * w1 % (d) w3 = v3 - (dot(w1,v3)/dot(w1,w1)) * w1 - (dot(w2,v3)/dot(w2,w2)) * w2 w4 = v4 - (dot(w1,v4)/dot(w1,w1)) * w1 - (dot(w2,v4)/dot(w2,w2)) * w2 - (dot(w3,v4)/dot(w3,w3)) * w3 % (e) u1 = w1 / sqrt(dot(w1,w1)) u2 = w2 / sqrt(dot(w2,w2)) u3 = w3 / sqrt(dot(w3,w3)) u4 = w4 / sqrt(dot(w4,w4)) % (f) Q = [u1 u2 u3 u4] Q.' * Q % (g) R = Q.' * A A - Q * R % (h) [Q1, R1] = qr(A,0) clear; % Problem 7 % (a) A = [ 5 9 -2 2 -1; 5 6 -8 7 -4; -5 -4 12 -9 6; 1 -1 -6 7 -3; 8 17 2 0 1; 8 15 -2 7 -1 ] rank(A) % (b) rref(A) % The 1st, 2nd and 4th columns form a basis for W % (c) B = [ 5 9 2; 5 6 7; -5 -4 -9; 1 -1 7; 8 17 0; 8 15 7 ] [Q, R] = qr(B,0) % The columns in Q give an orthonormal basis for W % (d) E = Q * Q.' v = [1; 1; 1; 1; 1; 1] E * v % (e) C = double(null(sym(B'))) rref(B.') % (f) [Q1, R1] = qr(C,0) % (g) F = Q1 * Q1.' E + F % E + F is the identity matrix
A =
1.0000 1.0000 1.0000 1.0000
1.0000 0.9950 0.9900 0.9851
1.0000 0.9801 0.9605 0.9414
1.0000 0.9553 0.9127 0.8719
ans =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
ans =
6.5176e-11
B =
1.0000 1.0000 1.0000 1.0000
1.0000 0.9801 0.9605 0.9414
1.0000 0.8776 0.7702 0.6759
1.0000 0.5403 0.2919 0.1577
ans =
1.7052e-05
A =
0 3 1 8 1
-9 -12 4 11 -2
-3 -5 1 1 -1
-6 -4 4 18 0
ans =
2
ans =
1 0 -8/9 -43/9 -2/9
0 1 1/3 8/3 1/3
0 0 0 0 0
0 0 0 0 0
v1 =
0
2
-4
-3
1
0
v2 =
1
2
-12
3
-2
2
v3 =
2
1
-3
-2
2
4
v4 =
2
-2
0
5
-1
4
A =
0 1 2 2
2 2 1 -2
-4 -12 -3 0
-3 3 -2 5
1 -2 2 -1
0 2 4 4
ans =
1 0 0 -9/5
0 1 0 2/5
0 0 1 4/5
0 0 0 0
0 0 0 0
0 0 0 0
P =
1 1 2 4
0 2 -1 -1
0 0 3 0
0 0 0 1
Q =
1 2 0 0
3 1 3 3
0 0 -1 0
1 0 4 0
R =
0 0 12 1
1/2 1/2 -5 3/2
0 0 -3 0
-1/6 1/2 -23/3 -11/6
Q1 =
0 0 4 1
1/2 0 -2 -1/2
0 0 -1 0
-1/6 1/3 -7/3 -5/6
ans =
1
-1/2
0
-5/6
v =
1
2
3
4
ans =
40
-15/2
-9
-59/2
ans =
4*t^3 + 9*t^2 - 3*t + 25
A =
7 -6 20
-18 15 -50
-8 7 -23
ans =
-3 8 -20
4 -17 40
2 -8 19
ans =
-5 -2 0
14 1 10
6 1 3
ans =
1 0 0
0 1 0
0 0 1
ans =
7 -6 20
-18 15 -50
-8 7 -23
ans =
-3 8 -20
4 -17 40
2 -8 19
ans =
-5 -2 0
14 1 10
6 1 3
ans =
1 0 0
0 1 0
0 0 1
P =
Columns 1 through 2
-361/1089 - 361/5445i -361/1089 + 361/5445i
2178/2527 + 0i 2178/2527 + 0i
3332/8817 + 71/10709i 3332/8817 - 71/10709i
Column 3
-881/2158 + 0i
881/1079 + 0i
881/2158 + 0i
D =
Columns 1 through 2
* + 1i 0 + 0i
0 + 0i * - 1i
0 + 0i 0 + 0i
Column 3
0 + 0i
0 + 0i
-1 + 0i
ans =
Columns 1 through 2
-1 + * 0 + 0i
0 + 0i -1 - *
0 + 0i 0 + 0i
Column 3
0 + 0i
0 + 0i
1 + 0i
ans =
Columns 1 through 2
* - 1i 0 + 0i
0 + 0i * + 1i
0 + 0i 0 + 0i
Column 3
0 + 0i
0 + 0i
-1 + 0i
ans =
Columns 1 through 2
1 - * 0 + 0i
0 + 0i 1 + *
0 + 0i 0 + 0i
Column 3
0 + 0i
0 + 0i
1 + 0i
v1 =
-14
13
7
0
3
v2 =
-7
-1
14
2
1
v3 =
-8
0
4
6
-3
v4 =
-3
15
-13
12
13
A =
-14 -7 -8 -3
13 -1 0 15
7 14 4 -13
0 2 6 12
3 1 -3 13
ans =
4
w1 =
-14
13
7
0
3
w2 =
-0.8440
-6.7163
10.9220
2.0000
-0.3191
w3 =
-3.3481
-1.5097
-2.2599
5.2507
-3.8095
w4 =
3.6937
1.3214
-0.7119
11.9857
13.1723
u1 =
-0.6807
0.6321
0.3404
0
0.1459
u2 =
-0.0649
-0.5163
0.8396
0.1537
-0.0245
u3 =
-0.4298
-0.1938
-0.2901
0.6741
-0.4891
u4 =
0.2024
0.0724
-0.0390
0.6568
0.7218
Q =
-0.6807 -0.0649 -0.4298 0.2024
0.6321 -0.5163 -0.1938 0.0724
0.3404 0.8396 -0.2901 -0.0390
0 0.1537 0.6741 0.6568
0.1459 -0.0245 -0.4891 0.7218
ans =
1.0000 0.0000 0.0000 -0.0000
0.0000 1.0000 0.0000 -0.0000
0.0000 0.0000 1.0000 0.0000
-0.0000 -0.0000 0.0000 1.0000
R =
20.5670 9.0436 6.3694 8.9950
-0.0000 13.0082 4.8736 -16.9391
0.0000 0.0000 7.7896 3.8849
-0.0000 -0.0000 0 18.2500
ans =
1.0e-14 *
0 0 0 -0.2220
0 0.2220 0.0222 0.3553
0.0888 0 0 0.1776
0.1643 0.0222 -0.1776 0.1776
0.1776 0.1998 0.0888 0.3553
Q1 =
-0.6807 -0.0649 -0.4298 -0.2024
0.6321 -0.5163 -0.1938 -0.0724
0.3404 0.8396 -0.2901 0.0390
0 0.1537 0.6741 -0.6568
0.1459 -0.0245 -0.4891 -0.7218
R1 =
20.5670 9.0436 6.3694 8.9950
0 13.0082 4.8736 -16.9391
0 0 7.7896 3.8849
0 0 0 -18.2500
A =
5 9 -2 2 -1
5 6 -8 7 -4
-5 -4 12 -9 6
1 -1 -6 7 -3
8 17 2 0 1
8 15 -2 7 -1
ans =
3
ans =
1 0 -4 0 -2
0 1 2 0 1
0 0 0 1 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
B =
5 9 2
5 6 7
-5 -4 -9
1 -1 7
8 17 0
8 15 7
Q =
-0.3501 0.0612 0.2657
-0.3501 -0.3741 0.1088
0.3501 0.6642 -0.3143
-0.0700 -0.3940 -0.5430
-0.5601 0.4751 0.2974
-0.5601 0.1849 -0.6600
R =
-14.2829 -24.5049 -10.7121
0 6.8927 -9.9380
0 0 -4.2996
E =
0.1969 0.1286 -0.1654 -0.1438 0.3041 0.0321
0.1286 0.2743 -0.4052 0.1128 0.0507 0.0551
-0.1654 -0.4052 0.6625 -0.1156 0.0260 0.1342
-0.1438 0.1128 -0.1156 0.4549 -0.3094 0.3247
0.3041 0.0507 0.0260 -0.3094 0.6279 0.2053
0.0321 0.0551 0.1342 0.3247 0.2053 0.7835
v =
1
1
1
1
1
1
ans =
0.3524
0.2164
0.1365
0.3236
0.9046
1.5348
C =
1.7000 -1.9000 0.6000
-3.3500 -0.5500 -5.8000
-1.4500 -0.8500 -3.6000
1.0000 0 0
0 1.0000 0
0 0 1.0000
ans =
1.0000 0 0 -1.7000 1.9000 -0.6000
0 1.0000 0 3.3500 0.5500 5.8000
0 0 1.0000 1.4500 0.8500 3.6000
Q1 =
-0.4097 -0.7940 -0.0689
0.8074 -0.2444 -0.1185
0.3495 -0.3636 -0.2883
-0.2410 0.0038 -0.6978
0 0.4213 -0.4412
0 0 0.4653
R1 =
-4.1491 0.0374 -6.1869
0 2.3735 2.2503
0 0 2.1491
F =
0.8031 -0.1286 0.1654 0.1438 -0.3041 -0.0321
-0.1286 0.7257 0.4052 -0.1128 -0.0507 -0.0551
0.1654 0.4052 0.3375 0.1156 -0.0260 -0.1342
0.1438 -0.1128 0.1156 0.5451 0.3094 -0.3247
-0.3041 -0.0507 -0.0260 0.3094 0.3721 -0.2053
-0.0321 -0.0551 -0.1342 -0.3247 -0.2053 0.2165
ans =
1.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000
-0.0000 1.0000 0.0000 -0.0000 0.0000 -0.0000
-0.0000 0.0000 1.0000 0.0000 0.0000 -0.0000
-0.0000 -0.0000 0.0000 1.0000 -0.0000 -0.0000
-0.0000 0.0000 0.0000 -0.0000 1.0000 -0.0000
-0.0000 -0.0000 -0.0000 -0.0000 -0.0000 1.0000