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+//page 282

+//Example 8.12

+clc;

+clear;

+close;

+b1 = [3 0 4];

+b2 = [-1 0 7];

+b3 = [2 9 11];

+disp(b1,'b1 = ');

+disp(b2,'b2 = ');

+disp(b3,'b3 = ');

+disp('Applying the Gram-Schmidt process to b1,b2,b3:');

+a1 = b1;

+disp(a1,'a1 = ');

+a2 = b2-((b2*b1')'/25*b1);

+disp(a2,'a2 = ');

+a3 = b3-((b3*b1')'/25*b1) - ((b3*a2')'/25*a2);

+disp(a3,'a3 = ');

+disp('{a1,a2,a3} are mutually orthogonal and hence forms orthogonal basis for R^3');

+disp('Any arbitrary vector {x1,x2,x3} in R^3 can be expressed as:');

+disp('y = {x1,x2,x3} = (3*x1 + 4*x3)/25*a1 + (-4*x1 + 3*x3)/25*a2 + x2/9*a3');

+x1 = 1;

+x2 = 2;

+x3 = 3;

+y = (3*x1 + 4*x3)/25*a1 + (-4*x1 + 3*x3)/25*a2 + x2/9*a3;

+disp(x1,'x1 = ');

+disp(x2,'x2 = ');

+disp(x3,'x3 = ');

+disp(y,'y = ');

+disp('i.e. y = [x1 x2 x3], according to above equation.');

+disp('Hence, we get the orthonormal basis as:');

+disp(',',1/5*a1);

+disp(',',1/5*a2);

+disp(1/9*a3);

+//end