[codeofthedamned]

Translation itself is not a linear transformation. To simply translate a vertex is a matrix addition(subtraction) problem.

The only reason that it is possible in the transformation matrix is because of the addition of the fourth parameter in a vertex [x y z 1] to create a homogenous linear system. This allows the translation to become a part of the transformation matrix.

[pandaman]

I am not sure what you mean. If by "matrix addition" you meant adding vectors then it's obvious but adding actual matrices won't give you any translation operators because no matter how many matrices you have added together you still end up with a matrix, which, indeed, can only represent linear operators.

[codeofthedamned]

I believe we are mincing words with the added confusion of two mathematical meanings for the word vertex in this discussion.

The single row matrix, called a vertex, and a Euclidean vertex which we use to represent the points in the geometry, and happen to store in a vertex (matrix).

I do agree with your statement.