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In linear transformations between abstract n-dimensional vector spaces the operators are always matrices

In linear transformations between or within abstract finite dimensional vector spaces the operators are always matrices. Why? Because of the role that matrices play in representing linear transformations in these spaces. Matrices containing the coordinate vectors of basis vectors are the means utilized to define linear transformations between two abstract finite dimensional spaces.

Because of this, when we talk about the sum of two linear transformations or the product of a linear transformation by a scalar, we are really talking about the sum of two matrices or the product of a scalar times a matrix.

Note that there is a conceptual difference between an operator and a linear mapping or linear transformation. An operator, such as a matrix, is conceived as an object. You can do operations on objects like matrices. A linear mapping or linear transformation is a functional assignment. It doesn't make sense to do operations such as addition on functional assignments.