Plücker coordinates: Difference between revisions
note about duality |
spacing nit |
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\begin{align} | \begin{align} | ||
\iota: \mathrm{Gr} (k, n) | \iota: \mathrm{Gr} (k, n) | ||
& \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\ | & \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\ | ||
\operatorname {span} (m_1, \ldots, m_k) | \operatorname {span} (m_1, \ldots, m_k) | ||
& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | & \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | ||
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</math> | </math> | ||
Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose. | Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose. | ||
While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>. | While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>. | ||