A linear representation is a mathematical concept that describes how a group or algebraic structure is represented by a set of linear transformations. It involves assigning a linear transformation to each element of the group or structure, preserving its algebraic operations.
Linear representations are widely used in physics and other sciences to study the symmetries of physical systems. They provide a powerful tool for analyzing and understanding the behavior of complex systems, such as molecules, crystals, and quantum particles.
There are two main types of linear representations: faithful and projective. A faithful representation is one in which each element of the group or structure is uniquely represented by a linear transformation. A projective representation, on the other hand, is one in which elements may have multiple representations.
A linear representation is a more general concept than a matrix representation. While a matrix representation assigns a matrix to each element of a group or structure, a linear representation can involve more general linear transformations, such as rotations, reflections, and translations.
Linear representations are closely related to symmetry because they describe how a group or structure can be transformed without changing its essential properties. In other words, linear representations capture the symmetries that exist in a system, making them a powerful tool for studying symmetrical structures and phenomena.