Can every linear transformation be considered a matrix transformation?
Not precisely: there's a vector space of linear transformations and a v.s. of matrices, and they both are isomorphic when properly taken the definition realm, the image and etc.
If we talka bout linear operators (from a v.s. to itself), then the v.s. of lin, transf. can be made into an algebra (multiplication of lin. transf. = function composition), which corresponds to square matrix multiplication and thus the isomorphism between them is one between algebras.