In the representation theory, a part of the group theory, the group representation describes abstract groups in terms of linear transformations of vector spaces; In particular, group representations can be used to represent group elements as matrices so that the group operation can be represented as a matrix multiplication.
Representations of groups are important because they allow to reduce many group-theoretical problems to problems in linear algebra, a part of mathematics that is well understood. They are also important in physics, for example because they describe how the symmetry group of a physical system influences the solutions of equations that describe the system.
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