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Symmetry group

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  Symmetry groups [ edit ] Main article:  Symmetry group See also:  Molecular symmetry ,  Space group ,  Point group , and  Symmetry in physics The (2,3,7) triangle group, a hyperbolic reflection group, acts on this  tiling  of the  hyperbolic  plane [50] Symmetry groups  are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). [51]  Conceptually, group theory can be thought of as the study of symmetry. [p]   Symmetries in mathematics  greatly simplify the study of  geometrical  or  analytical  objects. A group is said to  act  on another mathematical object  X  if every group element can be associated to some operation ...
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NORMED VECTOR SPACES AND INNER PRODUCT SPACES

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Normed vector spaces and inner product spaces [ edit ] Main articles:  Normed vector space  and  Inner product space "Measuring" vectors is done by specifying a  norm , a datum which measures lengths of vectors, or by an  inner product , which measures angles between vectors. Norms and inner products are denoted  | � |  and  ⟨ � , � ⟩ ,  respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm  | � | := ⟨ � , � ⟩ .  Vector spaces endowed with such data are known as  normed vector spaces  and  inner product spaces , respectively. [39] Coordinate space  � �  can be equipped with the standard  dot product : ⟨ � , � ⟩ = � ⋅ � = � 1 � 1 + ⋯ + � � � � . In  � 2 ,  this reflects the common notion of the angle between two vectors  �  and  � ,  by the  law of cosines : � ⋅ � = cos ⁡ ( ∠ ( � , � ) ) ⋅ | � | ⋅...
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