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Suppose we are asked to fit a straight line to the following data 1 3 -2 1 2 -1 so that mx + C= y Write the system of linear equations mx + c y. Determine the matrix A. and vector b to write this problem as a matrix equation A x b Describe the relationship between the vector b and the column space of (a) (b) A. Given that the matrix equation in part (a) has no solution, a least squares solution is sought by looking for a vector x that minimises the error e = Ax - b. This means that e is perpendicular to the columns of A. Show that this fact leads to the normal equations: (c) ATA X= ATb Use the matrix equation in part (c) to determine the associated normal equations and the least squares solution, x, for the matrix equation in part (a). (d)

Suppose we are asked to fit a straight line to the following data 1 3 -2 1 2 -1 so that mx + C= y Write the system of linear equations mx + c y. Determine the matrix A. and vector b to write this problem as a matrix equation A x b Describe the relationship between the vector b and the column space of (a) (b) A. Given that the matrix equation in part (a) has no solution, a least squares solution is sought by looking for a vector x that minimises the error e = Ax - b. This means that e is perpendicular to the columns of A. Show that this fact leads to the normal equations: (c) ATA X= ATb Use the matrix equation in part (c) to determine the associated normal equations and the least squares solution, x, for the matrix equation in part (a). (d)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 23EQ
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Suppose we are asked to fit a straight line to the following data m 2 1 3 -2 1 1 -1 1 so that mx + C = y Write the system of linear equations mx + c = y. Determine the matrix A. and vector b to write this problem as a matrix equation A x b Describe the relationship between the vector b and the column space of (a) (b) A. Given that the matrix equation in part (a) has no solution, a least squares solution is sought by looking for a vector x that minimises the error e = Ax - b. This means that e is perpendicular to the columns of A. Show that this fact leads to the normal equations: (c) ATA X = AT b Use the matrix equation in part (c) to determine the associated normal equations and the least squares solution, x, for the matrix equation in part (a). (d)
Transcribed Image Text:Suppose we are asked to fit a straight line to the following data m 2 1 3 -2 1 1 -1 1 so that mx + C = y Write the system of linear equations mx + c = y. Determine the matrix A. and vector b to write this problem as a matrix equation A x b Describe the relationship between the vector b and the column space of (a) (b) A. Given that the matrix equation in part (a) has no solution, a least squares solution is sought by looking for a vector x that minimises the error e = Ax - b. This means that e is perpendicular to the columns of A. Show that this fact leads to the normal equations: (c) ATA X = AT b Use the matrix equation in part (c) to determine the associated normal equations and the least squares solution, x, for the matrix equation in part (a). (d)
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