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Derivation of Gram-Schmidt orthogonalization procedure. Suppose *X _{1} , X_{2} , ..... ,
X_{m} * 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

1) Let _{ }*Y _{1} = X_{1}*

2) Let *Y _{2} = X_{2} + aY_{1} *.

Since *Y _{1}* and

*Y _{1}·Y_{2} = Y_{1}· (X_{2} + aY_{1} ) = Y_{1}·X_{2} + Y_{1 }· aY_{1 } = Y_{1} · X_{2} + a Y_{1}·Y_{1 }*

Consequently,

and

3) Let *Y _{3} = X_{3} + aY_{2 }+ bY_{1 .}*

Since *Y _{1}* ,

*Y _{1}·Y_{3} = Y_{1}· (X_{3} + aY_{2} + bY_{1} ) = Y_{1}·X_{3} + aY_{1 }· Y_{2 } + bY_{1 }· Y_{1 } = Y_{1} · X_{3} + b Y_{1}·Y_{1 }*

and

*Y _{2}·Y_{3} = Y_{2}· (X_{3} + aY_{2} + bY_{1} ) = Y_{2}·X_{3} + aY_{2}· Y_{2 } + bY_{2}· Y_{1 } = Y_{2} · X_{3} + a Y_{2}·Y_{2}*

Consequently,

and

4) Continue process until *Y _{m}* is obtained.

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