Plücker coordinates: Difference between revisions
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Give some trivial examples |
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Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power of our original space <math>\mathbb{R}^n</math>. The dimension of <math>\mathrm{Gr} (k, n)</math> is <math>k(n-k)</math>, while the dimension of <math>\Lambda^{k} \mathbb{R}^n</math> is <math>\binom{n}{k}</math>, which is typically much larger. | Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power of our original space <math>\mathbb{R}^n</math>. The dimension of <math>\mathrm{Gr} (k, n)</math> is <math>k(n-k)</math>, while the dimension of <math>\Lambda^{k} \mathbb{R}^n</math> is <math>\binom{n}{k}</math>, which is typically much larger. | ||
== | == Examples == | ||
The space of lines through the origin is exactly projective space, so <math>\mathrm{Gr} (1, n) \cong \mathbf{P} (\mathbb{R}^n)</math>. | |||
In 3 dimensions, a plane through the origin is completely defined by its normal, so we get that <math>\mathrm{Gr} (2, 3) \cong \mathrm{Gr} (1, 3) \cong \mathbf{P} (\mathbb{R}^3)</math>, the projective plane. | |||
The simplest non-trivial case is | The simplest non-trivial case is <math>\mathrm{Gr} (2, 4)</math>. | ||
An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix | An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix | ||
$$ | $$ | ||