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Exercise 6.2.6

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Take $X$ to be the space generated by $W$ and ${x}$ . Thus $X$ is also a finite-dimensional subspace. Apply Theorem 6.6 to $x$ in $X$ . We know that $x$ could be uniquely written as $u + v$ with $u \in W$ and $v \in V^{⊥}$ . Since $x \notin W$ , we have $v \neq 0$ . Pick $y = v$ . And we have

⟨ x, y ⟩ = ⟨ v, v ⟩ + ⟨ u, v ⟩ = ∥ v ∥^{2} > 0 .

jephianlin

2011-06-27 00:00

Exercise 6.2.6

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