Gram-Schmidt Orthogonalization/Scalar Product Space/Corollary
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Theorem
Let $\struct {V, q}$ be an $n$-dimensional scalar product space.
Let $V^*$ be the vector space dual to $V$.
Then there exists a basis $\tuple {\beta^i}$ for $V^*$ with respect to which $q$ has the expression:
- $q = \paren {\beta^1}^2 + \ldots + \paren {\beta^r}^2 - \paren {\beta^{r + 1}}^2 - \ldots - \paren {\beta^{r + s}}^2$
where:
- $r, s \in \N : r + s = n$.
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