Question

what are mertis of linear map?​

Answer 1

Answer:

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (for example, two vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. If a linear map is a bijection then it is called a linear isomorphism.

An important special case is when V = W, in which case a linear map is called a (linear) endomorphism of V. Sometimes the term linear operator refers to this case,[1] but the term "linear operator" can have different meanings different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W),[citation needed] or it can be used to emphasize that V is a function space, which is a common convention in functional analysis.[2] Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.

A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension);[3] for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory, it is a morphism in the category of modules over a given ring.