The concepts of linear algebra represent the end result of a series of generalizations and abstractions Motivation for extending the concept of a vector to include objects of an arbitrary nature. Generalization of the concept of a vector space Linear transformations, linear mappings, linear operators are linear vector functions assigning objects to objects Linear transformation, linear mapping. Operations, sum, product. Algebra of linear operators. Invertible operators. Isomorphisms and homomorphisms effected by linear transformations from one abstract vector space into another In linear transformations between abstract n-dimensional vector spaces the operators are always matrices Linear functional. Matrix representation. Dual space, conjugate space, adjoint space. Basis for dual space. Annihilator. Transpose of a linear mapping. Source.