linear operator eigenvalues

So I know how to handle a linear operator that Transform P3->P3. Essentially you mulitply by lamda and equate to the transform and group like terms (x^3,x^2, x, constant) and solve for lambdas. What if the operator transforms something in G3 where this is the 3D arrows (vector ijk). L(x)=T. Do you group ijk's instead of powers of x? Would you be solving Lambda*x=T? Just looking for a starting point for this problem. Once I have the eigenvalues I'm pretty sure I can determine the corresponding spaces.

Thanks