Problem 3. Let S be the shift operator on the set V of all real sequences, as in the Fibonacci notes. Let p(x)=(x)(x) be a quadratic polynomial with two different real roots =. Let L=p(S). We know L is a linear operator; let W=ker(L). 1. W is the solution space for a certain recurrence relation (also called a difference equation). Which one? 2. Show that (SI)(SI)=(SI)(SI). 3. In the Fibonacci notes we showed that certain eigenvectors of S were contained in W. Which ones, and why? 4. We also claimed, but did not prove, that W is spanned by eigenvectors of S. Since we have an explicit description of the eigenvectors of S, this gives us a formula for the elements of W. That. is, you've solved the recurrence relation: what are the solutions, explicitly? 5. The means to prove that W is spanned by eigenvectors of S is developed in the "even and odd functions" exercises (both versions). We need to show that any wW can be expressed as a sum of an eigenvector for S with eigenvalue and an eigenvector for S with eigenvalue . Think of this as like showing that a function is the sum of an even function (eigenvector for a certain operator for eigenvalue 1) and an odd function (eigenvector for that operator with eigenvalue -1 ). We define two linear operators on W, U(w)=(SI)(w)U(w)=(SI)(w) Show that U sends eigenvectors of S with eigenvalue to 0 . What does it do to eigenvectors of S with eigenvalue ? What can you say about U ? This may make it clear why we define two new operators, P=1U and P=1U. (What does this correspond to in the even and odd situation? What is there?) Show that PP=0 and that P2=P and P2=P. Show that P+P=I. Use this to