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+//page 60

+//Example 2.21

+clear;

+clc;

+close;

+a1 = [1 2 2 1];

+a2 = [0 2 0 1];

+a3 = [-2 0 -4 3];

+disp('Given row vectors are:');

+disp(a1,'a1 = ');

+disp(a2,'a2 = ');

+disp(a3,'a3 = ');

+disp('The matrix A from these vectors will be:');

+A = [a1; a2; a3];

+disp(A,'A = ');

+disp('Finding Row reduced echelon matrix of A that is given by R');

+disp('And applying same operations on identity matrix Q such that R = QA');

+Q = eye(3,3);

+disp(Q,'Q = ');

+T = A;              //Temporary matrix to store A

+disp('Applying row transformations on A and Q,we get');

+disp('R1 = R1-R2');

+A(1,:) = A(1,:) - A(2,:);

+Q(1,:) = Q(1,:) - Q(2,:);

+disp(A,'A = ');

+disp(Q,'Q = ');

+disp('R3 = R3 + 2*R1');

+A(3,:) = A(3,:) + 2*A(1,:);

+Q(3,:) = Q(3,:) + 2*Q(1,:);

+disp(A,'A = ');

+disp(Q,'Q = ');

+disp('R3 = R3/3');

+A(3,:) = 1/3*A(3,:);

+Q(3,:) = 1/3*Q(3,:);

+disp(A,'A = ');

+disp(Q,'Q = ');

+disp('R2 = R2/2');

+A(2,:) = 1/2*A(2,:);

+Q(2,:) = 1/2*Q(2,:);

+disp(A,'A = ');

+disp(Q,'Q = ');

+disp('R2 = R2 - 1/2*R3');

+A(2,:) = A(2,:) - 1/2*A(3,:);

+Q(2,:) = Q(2,:) - 1/2*Q(3,:);

+disp(A,'A = ');

+disp(Q,'Q = ');

+R = A;

+A = T;

+disp('Row reduced echelon matrix:');

+disp(R,'R = ');

+disp(Q,'Q =');

+//part a

+disp(rank(R),'rank of R = ');

+disp('Since, Rank of R is 3, so a1, a2, a3 are independent');

+//part b

+disp('Now, basis for W can be given by row vectors of R i.e. p1,p2,p3');

+disp('b is any vector in W. b = [b1 b2 b3 b4]');

+disp('Span of vectors p1,p2,p3 consist of vector b with b3 = 2*b1');

+disp('So,b = b1p1 + b2p2 + b4p3');

+disp('And,[p1 p2 p3] = R = Q*A');

+disp('So, b = [b1 b2 b3]* Q * A');

+disp('hence, b = x1a1 + x2a2 + x3a3 where x1 = [b1 b2 b4] * Q(1) and so on'); //Equation 1

+//part c

+disp('Now, given 3 vectors a1'' a2'' a3'':');

+c1 = [1 0 2 0];

+c2 = [0 2 0 1];

+c3 = [0 0 0 3];

+disp(c1,'a1'' = ');

+disp(c2,'a2'' = ');

+disp(c3,'a3'' = ');

+disp('Since a1'' a2'' a3'' are all of the form (y1 y2 y3 y4) with y3 = 2*y1, hence they are in W.');

+disp('So, they are independent.');

+//part d

+c = [c1; c2; c3];

+P = eye(3,3);

+for i = 1:3

+    b1 = c(i,1);

+    b2 = c(i,2);

+    b4 = c(i,4);

+    x1 = [b1 b2 b4] * Q(:,1);

+    x2 = [b1 b2 b4]*Q(:,2);

+    x3 = [b1 b2 b4]*Q(:,3);

+   P(:,i) = [x1; x2; x3];

+end

+disp('Required matrix P such that X = PX'' is:');

+disp(P,'P = ');

+//end