m235notes11a - Notes 11a: Markov Processes, Dimension,...

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Unformatted text preview: Notes 11a: Markov Processes, Dimension, Rank-Nullity Lecture Oct , 2011 Application 1. We are given a submersible self-propelled vehicle with three propulsion devices. Each of these is able to provide thrust in a single direction by any amount, positive or negative. The directions are given by the three vectors v 1 ,v 2 ,v 3 . By adjusting the amount of thrust of each device, can we achieve motion in any direction and at any speed we desire? To answer this we need to answer the question “Does the set { v 1 ,v 2 ,v 3 } span R 3 ?’ This is the same as asking “Can every vector in R 3 be written as a linear combination of v 1 ,v 2 ,v 3 ? Assume the answer to the questions is yes. For safety purposes we wish to add a ‘redundant’ propulsion device associated to a vector w . We desire that each of the sets { w,v 1 ,v 2 } , { w,v 1 ,v 3 }{ w,v 2 ,v 3 } span R 3 . Application 2. In one sense we have not gone far beyond the first algorithm in the course, Gauss elimination, an algorithm that solved systems of equations. We have tried to introduce a multitude of ways of looking at these systems of equations. We have introduced functions, functions defined by matrices, to recast these systems. We apply this idea of functions defined by matrices. We assume we have a community of 1000 shoppers and two grocery stores A,B . Each week each shopper goes to one of the stores and not the other. Each week 20% of the shoppers from grocery A switch to B and the rest return to A , while each week 30% of the shoppers at B switch to A and the rest return to B . Let x ,y denote the initial number of shoppers at A,B , so x + y = 1000 . Let x n ,y n denote the number of shoppers at A,B at week n . The behavior of the shoppers is . 8 x n + . 3 y n = x n +1 , . 2 x n + . 7 y n = y n +1 ....
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This note was uploaded on 11/22/2011 for the course MATH 235 taught by Professor Markman during the Fall '08 term at UMass (Amherst).

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m235notes11a - Notes 11a: Markov Processes, Dimension,...

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