"A linear map is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication."
A linear transformation is a function that maps vectors from one space to another. Linear transformations can be represented by matrices, such as rotation, reflection, and scaling.
"Linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function."
"The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism."
"If a linear map is a bijection, then it is called a linear isomorphism."
"In the case where V = W, a linear map is called a linear endomorphism."
"The term 'linear operator' can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces or that V is a function space."
"A linear map from V to W always maps the origin of V to the origin of W."
"It maps linear subspaces in V onto linear subspaces in W, possibly of a lower dimension."
"Yes, linear maps can often be represented as matrices."
"Simple examples include rotation and reflection linear transformations."
"Linear maps are the morphisms of vector spaces in the language of category theory."
"Sometimes the term linear function has the same meaning as linear map, while in analysis it does not."