> I always hated the dull mathematic definition of "a matrix represents a linear map" because what the matrix actually represents is completely context dependen

> I always hated the dull mathematic definition of "a matrix represents a linear map" because what the matrix actually represents is completely context dependent. The projection and translation matrices you mentioned represent restricted sets of linear maps in some vector space, but the meaning of the restriction is also captured in other mathematical structures. See affine spaces and projective spaces.