The operator is a reflection at the hyperplane orthogonal to for vectors in the plane spanned by and ; i. e. it acts as a reflection across.

A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This is a reflection in the hyperplane perpendicular to v ( negating any vector component parallel to v ).

# For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α.

In mathematics, a reflection ( also spelled reflexion ) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points ; this set is called the axis ( in dimension 2 ) or plane ( in dimension 3 ) of reflection.

Typically, however, unqualified use of the term " reflection " means reflection in a hyperplane.

Given a vector a in Euclidean space R < sup > n </ sup >, the formula for the reflection in the hyperplane through the origin, orthogonal to a, is given by

The formula for the reflection in the affine hyperplane not through the origin is

A Householder reflection ( or Householder transformation ) is a transformation that takes a vector and reflects it about some plane or hyperplane.

In linear algebra, a Householder transformation ( also known as Householder reflection or elementary reflector ) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.

The reflection of a point x about this hyperplane is:

The Householder Transformation is a reflection about a certain hyperplane, namely, the one with unit normal vector v, as stated earlier.

A reflection is a map that transforms an object into its mirror image with respect to a " mirror ", which is a hyperplane of fixed points in the geometry.

When working over finite fields, one defines a " reflection " as a map that fixes a hyperplane ( otherwise for example there would be no reflections in characteristic 2, as so reflections are the identity ).

The historian can only point out those lines which were major enough to find reflection in our limited evidence, and must hope that future excavations will enrich our understanding.

A sufficiently good approximation for determining the end reflection losses R can be obtained from the angle independent Fresnel formula: Af.

The global cooling effect of carbon sequestration by forests is partially counterbalanced in that reforestation can decrease the reflection of sunlight ( albedo ).

Light interaction is often approximated by the even simpler models: diffuse reflection and specular reflection, although both can be BRDFs.

It can be seen that the order of these operations does not matter ( the translation can come first, followed by the reflection ).

Language features include structured programming, reflection ( the ability to modify the program structure during program execution ), concatenative programming ( functions are composed with juxtaposition ) and extensibility ( the programmer can create new commands ).

The combined reflection coefficient for this case is 2R /( 1 + R ), when interference can be neglected ( see below ).

which can be found by setting z = 1 / 2 in the reflection or duplication formulas, by using the relation to the beta function given below with x = y = 1 / 2, or simply by making the substitution u = √ t in the integral definition of the gamma function, resulting in a Gaussian integral.

After sunset an increase in the height of the E layer maximum increases the range to which radio waves can travel by reflection from the layer.

The E < sub > s </ sub > layer ( sporadic E-layer ) is characterized by small, thin clouds of intense ionization, which can support reflection of radio waves, rarely up to 225 MHz.

It reflects medium-frequency radio waves, and because of this reflection, radio waves can be propagated beyond the horizon.

On the night side of the Earth, the solar wind drags the ionosphere further away, thereby greatly increasing the range which radio waves can travel by reflection, called skywave.

This can happen quite naturally through reflection or deliberate recall ( also known as recapitulation ), often dependent on the perceived importance of the material.

This reflection can be reduced, however, by being careful to use only the lowest wattage necessary for the lamp, and setting spacing between lights appropriately.

Interference reflection microscopy can be obtained by using the same elements used by DIC, but without the prisms.

DHM can operate both in reflection and transmission mode.

The balance between reflection and absorption determines how white or how gray they are, although surface tarnish can obscure such observations.

The Asian elephant can recognize its own reflection in a mirror

The laws of reflection and refraction can be derived from Fermat's principle which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Reflections can be divided into two types: specular reflection and diffuse reflection.

Glossy surfaces can give both specular and diffuse reflection.

An idea, let us say, may be roughly defined as a theme or topic with which our reflection may be concerned.

The reflection of radiation originating from the anode holder and reflected back to it by the surrounding metal surfaces should also be small because of the peculiar characteristic of the metal surfaces and of the specific geometry.

A formulaic element need not be held meaningless merely because it was selected with little conscious reflection.

It might be anticipated that multiple coatings will reduce end reflection losses even further.

The leader Montgomery envisages will need to discipline himself, lead a carefully regulated and orderly life, allow time for quiet thought and reflection, adapt decisions and plans to changing situations, be ruthless, particularly with inefficiency, and be honest and morally proper.

Being a dimensionless fraction, it may also be expressed as a percentage, and is measured on a scale from zero for no reflecting power of a perfectly black surface, to 1 for perfect reflection of a white surface.

In a culture that set a high value on oratory and public performances of all kinds, in which the production of books was very labor-intensive, the majority of the population was illiterate, and where those with the leisure to enjoy literary works also had slaves to read for them, written texts were more likely to be seen as scripts for recitation than as vehicles of silent reflection.

The goddess held a lance whose gilt tip could be seen as a reflection by crews on ships rounding Cape Sounion, and a giant shield on the left side, decorated by Mys with images of the fight between the Centaurs and the Lapiths.

To prevent muscle weakness and incisional hernias, the portion of abdominal wall exposed by reflection of the rectus abdominis muscle may be strengthened by a piece of surgical mesh placed over the defect and sutured in place.

Mechanisms such as reflection on actual or mental actions vis-à-vis alternative solutions to problems, tagging new concepts or solutions to symbols that help one recall and mentally manipulate them are just a few examples of how mechanisms of cognitive development may be used to facilitate learning.

However, should a solution be equivalent to its own 90 degree rotation ( as happens to one solution with five queens on a 5x5 board ) that fundamental solution will have only two variants ( itself and its reflection ).

Should a solution be equivalent to its own 180 degree rotation ( but not to its 90 degree rotation ) it will have four variants ( itself, its reflection, its 180 degree rotation and the reflection of that ).

It is not possible for a solution to be equivalent to its own reflection ( except at n = 1 ) because that would require two queens to be facing each other.

Generally, however, Enki seems to be a reflection of pre-patriarchal times, in which relations between the sexes were characterised by a situation of greater gender equality.

Moore argued that once arguments based on the naturalistic fallacy had been discarded, questions of intrinsic goodness could only be settled by appeal to what he ( following Sidgwick ) called " moral intuitions :" self-evident propositions which recommend themselves to moral reflection, but which are not susceptible to either direct proof or disproof ( PE § 45 ).