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MATH 224 LINEAR ALGEBRA Problem Set 5 Due August 11 in Tutorial 1. Find an explicit solution to the recursive relation: a n = a n - 1 + 3 a n - 2 given the initial conditions a 0 = 1, a 1 = 2 . 2. Let T : V → V be a linear operator over F . Recall that a subspace W of V is called T -cyclic if it is a T -invariant subspace of V with the property that it has a nonzero vector w ∈ W such that: W = span F { w ,T w ,...,T k w }, for some number k . Also recall the de±nition of an annihilator subspace with respect to a polynomial q ( x ) ∈ F [ x ] : K q ( x ) = { v ∈ V | q (T) v = 0}. Suppose that W is a T -cyclic subspace of V . Show that W ⊂ K p ( x ) for some polynomial p ( x ) . 3. Are the matrices A =   1 - 4 0 1 - 2 1 1 - 3 2   , B =   1 - 2 - 2 1 1 1 0 - 2 - 1   , conjugate over Q ? Show your reasoning. 4. Let T : V → V be a linear operator over the ±eld F 5 . Suppose A = [T] α α is a matrix representation of T with respect to some basis and c A ( x ) = (1 - x )( x - 3) 2 ( x 2 + 2) 2 . Write a list of all possible elementary divisors , invariant factors , and rational canonical forms that the matrix A can have.