That's one weird operator. It treats polynomials as vectors, but includes the powers of x? I could understand treating polynomials as vectors of coefficients, b

That's one weird operator. It treats polynomials as vectors, but includes the powers of x? I could understand treating polynomials as vectors of coefficients, but to treat them as vectors of mononomials is just plain weird. Ignoring the question of why on earth you'd want to do something like this it would be far more natural to point out a matrix can be considered a vector of linear operators by writing it as a tensor product of basis vectors and co-vectors (so a column-vector of row-vectors basically), where for a matrix M with rows r_i we have: (M x)_i = r_i * x = r_i(x) and then take the smallest logical step to make it nonlinear which would be: (M x)_i = f_i(x) for general functions f_i. Of course this is just the definition of a multivariate function, so perhaps it would be more interesting to look at some intermediate version where f_i is some linear combination of functions f_ij such that: (M x)_i = Sum_j f_ij(x_j) again analogous to how matrices work.