Equivalent Properties of Nondegenerate Subspace of Scalar Product Space
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Theorem
Let $\struct {V, q}$ be a scalar product space.
Let $S \subseteq V$ be a linear subspace.
Let $S^\perp$ be the vector subspace perpendicular to $S$ with respect to $q$.
Then the following are equivalent:
- $S$ is nondegenerate;
- $S^\perp$ is nondegenerate;
- $S \cap S^\perp = \set 0$
- $V = S \oplus S^\perp$
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