Linear Algebra/Topic: Linear Recurrences/Solutions

Solutions Problem 1 Solve each homogeneous linear recurrence relations. <li> <li> <li> Answer <li> We express the relation in matrix form. The characteristic equation of the matrix has roots of and . Any function of the form satisfies the recurrence. <li> This is like the prior part, but simpler. The matrix expression of the relation is and the characteristic equation of the matrix has the single root . Any function of the form satisfies this recurrence. <li> In matrix form the relation gives this characteristic equation. Problem 2exer:SolvePartRecurSoltn Give a formula for the relations of the prior exercise, with these initial conditions. <li> , <li> , <li> , , . Problem 3exer:SeqToRnLinMap Check that the isomorphism given between and is a linear map. It is argued above that this map is one-to-one. What is its inverse? Problem 4exer:CharEqnIsDeter Show that the characteristic equation of the matrix is as stated, that is, is the polynomial associated with the relation. (Hint: expanding down the final column, and using induction will work.) Problem 5exer:SoltnsLinRecur Given a homogeneous linear recurrence relation , let , ..., be the roots of the associated polynomial. <li> Prove that each function satisfies the recurrence (without initial conditions). <li> Prove that no is . <li> Prove that the set is linearly independent. Problem 6 (This refers to the value given in the computer code below.) Transferring one disk per second, how many years would it take the priests at the Tower of Hanoi to finish the job?