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(See Steps) Let x_1,x_2,...,x_n be a sequence of linearly independent vectors in an inner product space. Define the vectors inductively as e_1=(x_1)/(||x_1||)

Question: Let \({{x}_{1}},{{x}_{2}},...,{{x}_{n}}\) be a sequence of linearly independent vectors in an inner product space. Define the vectors inductively as

\[\begin{aligned} & {{e}_{1}}=\frac{{{x}_{1}}}{||{{x}_{1}}||} \\ & {{f}_{n}}={{x}_{n}}-\sum\limits_{j=1}^{n-1}{\left( {{x}_{n}},{{e}_{j}} \right){{e}_{j}},\,\,\,n\ge 2} \\ \end{aligned}\]

Show that \[\left\{ {{e}_{n}} \right\}_{n=1}^{\infty }\] is an orthonormal sequence having the same closed linear span as the x j ’s.

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Deliverable: Word Document


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