Derivation of Gram-Schmidt orthogonalization procedure. Suppose X1 , X2 , ..... , Xm constitute a basis of some vector space. We wish to devise a procedure for generating from these m vectors an orthogonal basis for the space. Denote by Y1 , Y2 , ..... , Ym the set of mutually orthogonal vectors to be found.
1) Let Y1 = X1
2) Let Y2 = X2 + aY1 .
Since Y1 and Y2 are to be mutually orthogonal we specify the condition
Y1·Y2 = Y1· (X2 + aY1 ) = Y1·X2 + Y1 · aY1 = Y1 · X2 + a Y1·Y1 = 0
3) Let Y3 = X3 + aY2 + bY1 .
Since Y1 , Y2 and Y3 are to be mutually orthogonal we specify the conditions
Y1·Y3 = Y1· (X3 + aY2 + bY1 ) = Y1·X3 + aY1 · Y2 + bY1 · Y1 = Y1 · X3 + b Y1·Y1 = 0
Y2·Y3 = Y2· (X3 + aY2 + bY1 ) = Y2·X3 + aY2· Y2 + bY2· Y1 = Y2 · X3 + a Y2·Y2 = 0
4) Continue process until Ym is obtained.