ABSTRACT
In this chapter, the author explains linear transformations, which are the most important functions between vector spaces. The modern and more precise definition of a vector space was introduced by Giuseppe Peano in 1888. By 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and the methods of previous centuries were generalized as the abstract algebra. However, for any particular linear transformation, the matrix will be determined by both the linear transformation and the choice of bases for the domain vector space and the range vector space. Hermann Grassmann introduces the notion of the linearly dependent vectors and develops the “elementary” theory of the finite-dimensional vector spaces, as can be found in all of today’s books on the linear algebra.
