(See Steps) The Gram-Schmidt process. Let x1, x2, ... be a sequence of linearly independent vectors in an inner product space. Define vectors inductively
Question: The Gram-Schmidt process. Let x1, x2, ... be a sequence of linearly independent vectors in an inner product space. Define vectors inductively as follows.
\({{e}_{1}}={{x}_{1}}/||{{x}_{1}}||\)
\[\begin{aligned}& {{f}_{n}}={{x}_{n}}-\sum\limits_{j=1}^{n-1}{\left( {{x}_{n}},{{e}_{j}} \right){{e}_{j}}},\,\,\,\,n\ge 2 \\
& {{e}_{n}}={{f}_{n}}/||{{f}_{n}}||,\,\,\,n\ge 2 \\
\end{aligned}\]
Show that \[\left( {{e}_{n}} \right)_{1}^{\infty }\] is an orthonormal sequence having the same closed linear span as the xj’s.
Deliverable: Word Document 
Solution: The first three Legendre polynomials are P_0(x)=1,
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[See Steps] Let (e_n)_1^∞ be a complete orthonormal sequence
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