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Exercise 6.2.8

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We apply induction on $n$ . When $n = 1$ , the Gram-Schmidt process always preserve the first vector. Suppose the statement holds for $n \leq k$ . Consider the a orthogonal set of nonzero vectors

{w_{1}, w_{2}, \dots, w_{k}} .

By induction hypothesis, we know that the vectors $v_{i} = w_{i}$ for $i = 1, 2, \dots, k - 1$ , where $v_{i}$ is the vector derived from the process. Now we apply the process the find

v_{n} = w_{n} - \sum_{i = 1}^{k - 1} \frac{⟨ w_{n}, v_{i} ⟩}{∥ v_{i} ∥^{2}} v_{i}

= w_{n} - 0 = w_{n} .

So we get the desired result.

jephianlin

2011-06-27 00:00

Exercise 6.2.8

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