I rather dislike this way of saying things. A matrix is a matrix, its a table of numbers, and one can usefully define operations like addition and multiplication on them.
A linear map is a linear map, it maps between vector spaces and it obeys some nice axioms. You can define addition and composition as operations on them.
It is a quite interesting and non-trivial theorem that if you fix a particular choice of basis then you get "for free" a bijection between linear maps and matrices. The bijection between matrices and linear maps is completely dependent on the basis you choose, however, and there certainly isn't a canonical way to choose the basis.
Often it is natural to change basis to make it easier solve some particular problem, and then the matrix that represents a particular linear map will change, but the properties of the linear map won't change (for example its rank, kernel, eigenvectors/values etc).
A linear map is a linear map, it maps between vector spaces and it obeys some nice axioms. You can define addition and composition as operations on them.
It is a quite interesting and non-trivial theorem that if you fix a particular choice of basis then you get "for free" a bijection between linear maps and matrices. The bijection between matrices and linear maps is completely dependent on the basis you choose, however, and there certainly isn't a canonical way to choose the basis.
Often it is natural to change basis to make it easier solve some particular problem, and then the matrix that represents a particular linear map will change, but the properties of the linear map won't change (for example its rank, kernel, eigenvectors/values etc).