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P23.2. The population of Canada, as determined by the Canadian census, was as follows: year 2001 2006 2011 2016 2021 population (in millions) 30.0 31.6 33.5 35.2 37.0 Let t denote the time measured in years from 2001. (i) Suppose that population of Canada (measure in millions) is modeled by the linear function pe(t) = mt+b. Find the least-squares estimates for the parameters m and b. (ii) Suppose that the population of Canada (measure in millions) is modeled by the exponential function pe(t) =ce, Linearize the model and use the least-squares method to estimate the parameters c and 2

P23.2. The population of Canada, as determined by the Canadian census, was as follows: year 2001 2006 2011 2016 2021 population (in millions) 30.0 31.6 33.5 35.2 37.0 Let t denote the time measured in years from 2001. (i) Suppose that population of Canada (measure in millions) is modeled by the linear function pe(t) = mt+b. Find the least-squares estimates for the parameters m and b. (ii) Suppose that the population of Canada (measure in millions) is modeled by the exponential function pe(t) =ce, Linearize the model and use the least-squares method to estimate the parameters c and 2

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 34EQ
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(please solve question 23.2 within 15 minutes I will give thumbs up)
P23.1. Consider To 1 1 A = and b3= 1 01 0 11 Show that a least-squares solution to Ax = b is not unique and solve the normal equations to find all of the least-squares solutions. P23.2. The population of Canada, as determined by the Canadian census, was as follows: year 2001 2006 2011 2016 2021 population (in millions) 30.0 31.6 33.5 35.2 37.0 Let t denote the time measured in years from 2001. (1) Suppose that population of Canada (measure in millions) is modeled by the linear function pe(t) = mt +b. Find the least-squares estimates for the parameters m and b. (ii) Suppose that the population of Canada (measure in millions) is modeled by the exponential function pe(t) = ce, Linearize the model and use the least-squares method to estimate the parameters c and 2. P23.3. Fix a nonnegative integer n. Consider the finite set 27(n–1) ={0. } = {# eR|0<{<n=1} X:= 2n 4x of n real numbers corresponding the left endpoints of equally spaced intervals between 0 and 2z. Let V=C the complex inner product space, consisting of all functions from the finite set X to C equipped with the inner product n-1 (Fi18) := E f(x)g(x) =Ef)8(#). %3D XEX l=0 (i) For all integers j satisfying 0<j<n-1, show that the functions w,(x)= exp(-jxi) are pairwise orthogonal and compute ||w,(x)|| - (ii) For all integers k satisfying 0 <k <n-1, consider the indicator function 1 ifx= 2nk 0ifx# %3D he (x) := Which function in the linear subspace W= Span (wo(x), w1(x),.., Wn-1(x)) CV is the best approximation the function h (x)?
Transcribed Image Text:P23.1. Consider To 1 1 A = and b3= 1 01 0 11 Show that a least-squares solution to Ax = b is not unique and solve the normal equations to find all of the least-squares solutions. P23.2. The population of Canada, as determined by the Canadian census, was as follows: year 2001 2006 2011 2016 2021 population (in millions) 30.0 31.6 33.5 35.2 37.0 Let t denote the time measured in years from 2001. (1) Suppose that population of Canada (measure in millions) is modeled by the linear function pe(t) = mt +b. Find the least-squares estimates for the parameters m and b. (ii) Suppose that the population of Canada (measure in millions) is modeled by the exponential function pe(t) = ce, Linearize the model and use the least-squares method to estimate the parameters c and 2. P23.3. Fix a nonnegative integer n. Consider the finite set 27(n–1) ={0. } = {# eR|0<{<n=1} X:= 2n 4x of n real numbers corresponding the left endpoints of equally spaced intervals between 0 and 2z. Let V=C the complex inner product space, consisting of all functions from the finite set X to C equipped with the inner product n-1 (Fi18) := E f(x)g(x) =Ef)8(#). %3D XEX l=0 (i) For all integers j satisfying 0<j<n-1, show that the functions w,(x)= exp(-jxi) are pairwise orthogonal and compute ||w,(x)|| - (ii) For all integers k satisfying 0 <k <n-1, consider the indicator function 1 ifx= 2nk 0ifx# %3D he (x) := Which function in the linear subspace W= Span (wo(x), w1(x),.., Wn-1(x)) CV is the best approximation the function h (x)?
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