Reflection #
A linear isometry equivalence K.reflection : E โโแตข[๐] E
in constructed, by choosing
for each u : E
, K.reflection u = 2 โข K.starProjection u - u
.
Auxiliary definition for reflection
: the reflection as a linear equivalence.
Equations
- K.reflectionLinearEquiv = LinearEquiv.ofInvolutive (2 โข โK.starProjection - LinearMap.id) โฏ
Instances For
Reflection in a complete subspace of an inner product space. The word "reflection" is sometimes understood to mean specifically reflection in a codimension-one subspace, and sometimes more generally to cover operations such as reflection in a point. The definition here, of reflection in a subspace, is a more general sense of the word that includes both those common cases.
Equations
- K.reflection = { toLinearEquiv := K.reflectionLinearEquiv, norm_map' := โฏ }
Instances For
The result of reflecting.
Reflection is its own inverse.
Reflection is its own inverse.
Reflecting twice in the same subspace.
Reflection is involutive.
Reflection is involutive.
Reflection is involutive.
A point is its own reflection if and only if it is in the subspace.
Reflection in the Submodule.map
of a subspace.
Reflection in the Submodule.map
of a subspace.
Reflection through the trivial subspace {0} is just negation.
The reflection in K
of an element of Kแฎ
is its negation.
The reflection in Kแฎ
of an element of K
is its negation.
The reflection in (๐ โ v)แฎ
of v
is -v
.