Gram-Schmidt Process, Orthogonalization Algorithm - Linear Algebra
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(Gram-Schmidt orthogonalization) We know from elementary linear algebra that any set of linearly independent vectors may be turned into a set of orthogonal vectors by the Gram-Schmidt orthogonalization process. The same process works for functions in L^2[a, b]. Let f1, f2, f3,... be an independent set of functions in L^2. Define the set of functions gn by $g1 = f1;g2 = f2 - ((f2, g1))/(g1^2) g1;g3 = f3 - ((f3, g2))/(g2^2) g2 - ((f3, g1))/(g1^2) g1,...thatgnis an orthogonal set.
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