Module 2 : Fundamentals of Vector Spaces
Section 5 : Gram-Schmidt Process and Orthogonal Polynomials
 
5. Gram-Schmidt Process and Orthogonal Polynomials
Given any linearly independent set in an inner product space, it is possible to construct an orthonormal set. This procedure is called Gram-Schmidt procedure. Consider a linearly independent set of vectors MATHin a inner product space we define $\QTR{bf}{e}^{(1)}$ as
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We form unit vector $\QTR{bf}{e}^{(2)}$ in two steps.
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whereMATHis component of $\QTR{bf}{x}^{(2)}$ along $\QTR{bf}{e}^{(1)}.$
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$.$By direct calculation it can be verified that MATH The remaining orthonormal vectors $\QTR{bf}{e}^{(i)}$ are defined by induction. The vector $\QTR{bf}{z}^{(k)}$ is formed according to the equation
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and
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It can be verified by direct computation that MATH for all $j<k$ as follows
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