Properties of Linear Subspace of Finite Dimensional Scalar Product Space

Theorem

Let $\struct {V, q}$ be a scalar product space.

Let $S \subseteq V$ be a linear subspace.

Then:

$\dim S + \dim S^\perp = \dim V$

$\paren {S^\perp}^\perp = S$

where $\dim$ denotes the dimension of vector space, and $S^\perp$ denotes the vector subspace perpendicular to $S$ with respect to $q$.

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