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+//page 301

+//Example 8.23

+clc;

+clear;

+close;

+disp('Linear transformation from V into W i.e. T is:');

+disp('T(x1,x2,x3) = ');

+disp('0    -x3    x2');

+disp('x3     0   -x1');

+disp('-x2   x1    0');

+disp('Then, T maps V onto W');

+disp('And, putting:');

+disp('A = ');

+disp('0    -x3    x2');

+disp('x3     0   -x1');

+disp('-x2   x1    0');

+disp('B = ');

+disp('0    -y3    y2');

+disp('y3     0   -y1');

+disp('-y2   y1    0');

+disp('we get,');

+disp('tr(AB'') = x3*y3 + x2*y2 + x1*y1 + x3*y3 + x2*y2 + x1*y1');

+disp('tr(AB'') = 2*(x1*y1 + x2*y2 + x3*y3)');

+disp('Thus, (a|b) = (Ta|Tb)');

+disp('T is vector space isomorphism');

+disp('T contains the standard and orthonormal basis consisting of matrices A1,A2,A3');

+A1 = [0 0 0;0 0 -1;0 1 0];

+A2 = [0 0 1;0 0 0;-1 0 0];

+A3 = [0 -1 0;1 0 0;0 0 0];

+disp(A1,'A1 = ');

+disp(A2,'A2 = ');

+disp(A3,'A3 = ');

+//end