WINTER 2007 assign8

# WINTER 2007 assign8 - Math 235 Assignment 8 Due 9:15am...

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Math 235 Assignment 8 Due 9:15am, Wednesday March 21, 2007. 1. From the Text § 5.4 #6 and determine the rank of T . Is T injective? Is T surjective? § 7.1 #24. 2. Consider the following linear recurrence equation: x n - 4 x n - 1 + 5 x n - 2 = 0 for n 2 , with initial conditions x 0 = 1 and x 1 = 1 . In this question you are asked to ﬁnd x n as a function of n through linear algebra by taking the following steps. No credit will be given for any other approach. (a) Let v n = ( x n , x n - 1 ) t for n 1 (so v n is a column vector). Express the recurrence equation in the form v n = Av n - 1 for n 2 , where A is a 2 × 2 matrix. Hence, express v n in terms of A and v 1 . (b) Find an expression for A N of the form a ( N ) A + b ( N ) I where N 2 is an integer, a ( N ) and b ( N ) are functions of N, and I is the 2 × 2 identity matrix. (c) Hence express x n explicitly in terms of n alone.

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WINTER 2007 assign8 - Math 235 Assignment 8 Due 9:15am...

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