Gram-Schmidt Orthogonalization/Scalar Product Space

Theorem

Let $\struct {V, q}$ be an $n$-dimensional scalar product space

Suppose $\tuple {v_i}$ is a nondegenerate basis for $V$.

Then there is an orthonormal basis $\tuple {b_i}$ such that:

$\ds \forall k \in \N_{> 0} : k \le n : \map \span {b_1, \ldots b_k} = \map \span {v_1, \ldots v_k}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics