Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 9E from Chapter 6.6 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 9E
Chapter:
Problem:
A certain experiment produces the data . Describe the model that produces a least-squares fit of these points by a function of the form
Step-by-Step Solution
Given Information
Given that a certain experiment produces the data $( 1,7.9 ) , ( 2,5.4 ) ,$ and $( 3 , - .9 ) .$ We have to describe the model that produces a least-squares fit given by: \[ y = A \cos x + B \sin x \]
Step-1: The set of equations
The equation at each point is \[\begin{array}{l} 7.9 = A\cos 1 + B\sin 1 + {\varepsilon _1}\\ 5.4 = A\cos 2 + B\sin 2 + {\varepsilon _2}\\ - 0.9 = A\cos 3 + B\sin 3 + {\varepsilon _3} \end{array}\] Write the system of equations $y = X \beta + \varepsilon$ \[ \left[ \begin{array} { c } { 7.9 } \\ { 5.4 } \\ { - 0.9 } \end{array} \right] = \left[ \begin{array} { c c } { \cos 1 } & { \sin 1 } \\ { \cos 2 } & { \sin 2 } \\ { \cos 3 } & { \sin 3 } \end{array} \right] \quad \left[ \begin{array} { c } { A } \\ { B } \end{array} \right] + \left[ \begin{array} { c } { \varepsilon _ { 1 } } \\ { \varepsilon _ { 2 } } \\ { \varepsilon _ { 3 } } \end{array} \right] \] Therefore, the design matrix is:
\[ X = \left[ \begin{array} { l l } { \cos 1 } & { \sin 1 } \\ { \cos 2 } & { \sin 2 } \\ { \cos 3 } & { \sin 3 } \end{array} \right] \]
Step-2: The least squares model
The required model is: \[ y = X \beta + \varepsilon \] with
\[\begin{array}{l} {\bf{y}} = \left[ {\begin{array}{*{20}{c}} {7.9}\\ {5.4}\\ { - 0.9} \end{array}} \right]\\ X = \left[ {\begin{array}{*{20}{l}} {\cos 1}&{\sin 1}\\ {\cos 2}&{\sin 2}\\ {\cos 3}&{\sin 3} \end{array}} \right]\\ \beta = \left[ {\begin{array}{*{20}{l}} A\\ B \end{array}} \right]\\ \varepsilon = \left[ {\begin{array}{*{20}{l}} {{\varepsilon _1}}\\ {{\varepsilon _2}}\\ {{\varepsilon _3}} \end{array}} \right] \end{array}\]
Given that a certain experiment produces the data $( 1,7.9 ) , ( 2,5.4 ) ,$ and $( 3 , - .9 ) .$ We have to describe the model that produces a least-squares fit given by: \[ y = A \cos x + B \sin x \]
Step-1: The set of equations
The equation at each point is \[\begin{array}{l} 7.9 = A\cos 1 + B\sin 1 + {\varepsilon _1}\\ 5.4 = A\cos 2 + B\sin 2 + {\varepsilon _2}\\ - 0.9 = A\cos 3 + B\sin 3 + {\varepsilon _3} \end{array}\] Write the system of equations $y = X \beta + \varepsilon$ \[ \left[ \begin{array} { c } { 7.9 } \\ { 5.4 } \\ { - 0.9 } \end{array} \right] = \left[ \begin{array} { c c } { \cos 1 } & { \sin 1 } \\ { \cos 2 } & { \sin 2 } \\ { \cos 3 } & { \sin 3 } \end{array} \right] \quad \left[ \begin{array} { c } { A } \\ { B } \end{array} \right] + \left[ \begin{array} { c } { \varepsilon _ { 1 } } \\ { \varepsilon _ { 2 } } \\ { \varepsilon _ { 3 } } \end{array} \right] \] Therefore, the design matrix is:
\[ X = \left[ \begin{array} { l l } { \cos 1 } & { \sin 1 } \\ { \cos 2 } & { \sin 2 } \\ { \cos 3 } & { \sin 3 } \end{array} \right] \]
Step-2: The least squares model
The required model is: \[ y = X \beta + \varepsilon \] with
\[\begin{array}{l} {\bf{y}} = \left[ {\begin{array}{*{20}{c}} {7.9}\\ {5.4}\\ { - 0.9} \end{array}} \right]\\ X = \left[ {\begin{array}{*{20}{l}} {\cos 1}&{\sin 1}\\ {\cos 2}&{\sin 2}\\ {\cos 3}&{\sin 3} \end{array}} \right]\\ \beta = \left[ {\begin{array}{*{20}{l}} A\\ B \end{array}} \right]\\ \varepsilon = \left[ {\begin{array}{*{20}{l}} {{\varepsilon _1}}\\ {{\varepsilon _2}}\\ {{\varepsilon _3}} \end{array}} \right] \end{array}\]