Differential Operators as Matrices
Preliminaries
1. Prove that P_2(R), the space of degree-2 polynomials with real number coefficients, is a vector space of R.
2. Prove that P_2(R) is isomorphic to R^3 as vector spaces over R. Note: With such isomorphism, we can consider polynomials to be vectors in R^3, and linear differential operators on P_2(x) as linear transformations of R^3.
3. Prove that {1, x, x^2} is a basis for P_2(R). That is, show that every element of P_2(R) can be written as a linear combination of these basis vectors.
4. Using the basis from prompt 3, find the matrix form of the linear differential operator d/dx. Note: The columns of your matrix will be the components of the basis elements. I recommend using the first column as the components of the image of 1, the second as the components of the image of x, and the third as the components of the image of x^2.
5. Check your answer to prompt 4 using the applet below. Note that your differential operator should take the form 0*d^2/dx^2 + 1*d/dx + 0*1.
6. Start playing around with differential operators of the form T = c_2D^2 + c_1D^1+ c_0D^0, where c_i are numbers, and D^i is the ith order derivative, by typing on coefficients. Can you determine the matrix form for a general linear differential operator of this form when it acts on an arbitrary polynomial in P_2(x)?