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Exercise 6.5.11

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Extend ${(\frac{1}{3}, \frac{2}{3}, \frac{2}{3})}$ to be a basis and do the Gram-Schmidt process to get an orthonormal basis. The extended basis could be

{(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}), (0, 1, 0), (0, 0, 1)}

and the orthonormal basis would be

{(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}), (- \frac{2}{3 \sqrt{5}}, \frac{5}{3 \sqrt{5}}, - \frac{2^{2}}{3 \sqrt{5}}), (- \frac{2}{\sqrt{5}}, 0, \frac{1}{\sqrt{5}})} .

So the matrix could be

(\begin{matrix} \frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\ - \frac{2}{3 \sqrt{5}} & \frac{5}{3 \sqrt{5}} & - \frac{2^{2}}{3 \sqrt{5}} \\ - \frac{2}{\sqrt{5}} & 0 & \frac{1}{\sqrt{5}} \end{matrix}) .

jephianlin

2011-06-27 00:00

Exercise 6.5.11

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