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1. Suppose that at some initial point in time 100,000 people live in Auckland city and 25,000 people live in its suburbs. After conducting a research, the Regional Planning Commission determines that the model governing people's movement between Auckland and its suburbs is given by the equations: an+1=0.95a +0.03s, 8n+1=0.05a, +0.978, where an is the number of people in Auckland and s, is the number of people in the suburbs. (a) Using column vectors, write the transition matrix A. (b) Assuming that the total population remains constant, by using Xn+1 = Ax,, to compute the first five state vectors, find the population of Auckland city and its suburbs over a period of five years. (e) Write x₁, as a linear combination of the eigenvectors of A and use it to predict the first five state vectors. Confirm that your answers are the same as in Question (1b). (d) What does the model predict in the long term. Note: we have used Xn to denote the vector an S

1. Suppose that at some initial point in time 100,000 people live in Auckland city and 25,000 people live in its suburbs. After conducting a research, the Regional Planning Commission determines that the model governing people's movement between Auckland and its suburbs is given by the equations: an+1=0.95a +0.03s, 8n+1=0.05a, +0.978, where an is the number of people in Auckland and s, is the number of people in the suburbs. (a) Using column vectors, write the transition matrix A. (b) Assuming that the total population remains constant, by using Xn+1 = Ax,, to compute the first five state vectors, find the population of Auckland city and its suburbs over a period of five years. (e) Write x₁, as a linear combination of the eigenvectors of A and use it to predict the first five state vectors. Confirm that your answers are the same as in Question (1b). (d) What does the model predict in the long term. Note: we have used Xn to denote the vector an S

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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1. Suppose that at some initial point in time 100,000 people live in Auckland city and 25,000 people live in its suburbs. After conducting a research, the Regional Planning Commission determines that the model governing people's movement between Auckland and its suburbs is given by the equations: an+1 = 0.95a, + 0.03s Sn+1 = 0.05a, +0.97sn where an, is the number of people in Auckland and s, is the number of people in the suburbs. (a) Using column vectors, write the transition matrix A. (b) Assuming that the total population remains constant, by using Xn+1 = Ax, to compute the first five state vectors, find the population of Auckland city and its suburbs over a period of five years. (e) Write x₁, as a linear combination of the eigenvectors of A and use it to predict the first five state vectors. Confirm that your answers are the same as in Question (1b). (d) What does the model predict in the long term. Note: we have used Xn to denote the vector an 8p
Transcribed Image Text:1. Suppose that at some initial point in time 100,000 people live in Auckland city and 25,000 people live in its suburbs. After conducting a research, the Regional Planning Commission determines that the model governing people's movement between Auckland and its suburbs is given by the equations: an+1 = 0.95a, + 0.03s Sn+1 = 0.05a, +0.97sn where an, is the number of people in Auckland and s, is the number of people in the suburbs. (a) Using column vectors, write the transition matrix A. (b) Assuming that the total population remains constant, by using Xn+1 = Ax, to compute the first five state vectors, find the population of Auckland city and its suburbs over a period of five years. (e) Write x₁, as a linear combination of the eigenvectors of A and use it to predict the first five state vectors. Confirm that your answers are the same as in Question (1b). (d) What does the model predict in the long term. Note: we have used Xn to denote the vector an 8p
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