Hodge dual: Difference between revisions
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== Definition == | == Definition == | ||
Given a rank ''k'' temperament on a JI subgroup of dimension ''n'', the mapping M can be written as a <math>k \times n</math> matrix. | Given a rank ''k'' [[abstract temperament]] on a [[JI subgroup]] of dimension ''n'', the [[mapping]] M can be written as a <math>k \times n</math> matrix. | ||
Writing the rows of this matrix as <math>v_1, \ldots, v_k</math>, the Plücker coordinates are <math>[v_1 \wedge \cdots \wedge v_k]</math>. | Writing the rows of this matrix as <math>v_1, \ldots, v_k</math>, the Plücker coordinates are <math>[v_1 \wedge \cdots \wedge v_k]</math>. | ||
This matrix can also be though of as representing a ''k''-plane, spanned by the rows. | This matrix can also be though of as representing a ''k''-plane, spanned by the rows. | ||
The kernel <math>\ker M</math>, representing the comma space, is an <math>(n - k)</math>-dimensional subspace of <math> \mathbb{R}^n </math>. | The kernel <math>\ker M</math>, representing the [[comma basis|comma space]], is an <math>(n - k)</math>-dimensional subspace of <math> \mathbb{R}^n </math>. | ||
Similarly, if we represent <math>\ker M</math> by a matrix K, then its Plücker coordinates are <math>[w_1 \wedge \cdots \wedge w_{n - k}]</math>, where \( w_i \) are the columns of K. | Similarly, if we represent <math>\ker M</math> by a matrix K, then its Plücker coordinates are <math>[w_1 \wedge \cdots \wedge w_{n - k}]</math>, where \( w_i \) are the columns of K. | ||