How Do You Spell LINEAR GROUP?

1. A linear group is a mathematical group consisting of linear transformations that map a vector space onto itself. In other words, it is a set of functions or operations that preserve the linear structure of a vector space.

   To understand this definition, it is necessary to clarify a few terms. A vector space is a mathematical structure that consists of a set of objects, known as vectors, along with a field of scalars (usually real or complex numbers) and specific operations for vector addition and scalar multiplication. Linear transformations, on the other hand, are functions that preserve vector addition and scalar multiplication properties of vector spaces. They can be represented by matrices and are commonly used to describe operations such as rotations, reflections, and dilations.

   A linear group, therefore, encompasses a collection of such linear transformations. For a group to be considered linear, it must fulfill certain properties. Firstly, it must contain an identity element, which is a transformation that leaves every vector unchanged. Secondly, it must be closed under composition, meaning that the composition of any two linear transformations in the group is also a linear transformation. Lastly, every linear transformation in the group must have an inverse, which is another linear transformation that, when composed, yields the identity transformation.

   Linear groups have significant applications in various branches of mathematics and physics. They often arise in the study of symmetry and the description of motion and transformations in vector spaces.

The word "linear" in "linear group" is derived from the Latin word "linea", meaning "line". In mathematics, a linear group refers to a group of linear transformations, which are functions that preserve straight lines. The term was first introduced by the German mathematician Sophus Lie in the late 19th century.