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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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