Find the least-squares solution to the equation 1 0 2 x = 1 2.1. Suppose = (x₁, x₂), then X1 x2
Find the least-squares solution to the equation 1 0 2 x = 1 2.1. Suppose = (x₁, x₂), then X1 x2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![# Least-Squares Solution to a System of Linear Equations
**Problem Statement:**
Find the least-squares solution \( \hat{x} \) to the equation
\[
\begin{bmatrix}
1 & 1 \\
0 & 2 \\
1 & 1
\end{bmatrix}
\hat{x} =
\begin{bmatrix}
2 \\
2 \\
-4
\end{bmatrix}.
\]
**Solution Steps:**
Suppose \( \hat{x} = (x_1, x_2) \), then we need to find the values of \( x_1 \) and \( x_2 \).
The system of equations, expressed in matrix form, is:
\[
\begin{bmatrix}
1 & 1 \\
0 & 2 \\
1 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
2 \\
2 \\
-4
\end{bmatrix}.
\]
By solving this system using the least-squares method, we can determine \( x_1 \) and \( x_2 \).
**Interactive Section:**
Please fill in the following to find the solution:
\[
x_1 = \_\_\_\_\_\_\_\_\_ ,
\]
\[
x_2 = \_\_\_\_\_\_\_\_\_ .
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F238584dd-2c47-447e-aaba-b9fce1ec1b5c%2F3e676b99-e3c3-412f-95ac-2124245cd89b%2Fip7dh2i_processed.png&w=3840&q=75)
Transcribed Image Text:# Least-Squares Solution to a System of Linear Equations
**Problem Statement:**
Find the least-squares solution \( \hat{x} \) to the equation
\[
\begin{bmatrix}
1 & 1 \\
0 & 2 \\
1 & 1
\end{bmatrix}
\hat{x} =
\begin{bmatrix}
2 \\
2 \\
-4
\end{bmatrix}.
\]
**Solution Steps:**
Suppose \( \hat{x} = (x_1, x_2) \), then we need to find the values of \( x_1 \) and \( x_2 \).
The system of equations, expressed in matrix form, is:
\[
\begin{bmatrix}
1 & 1 \\
0 & 2 \\
1 & 1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}
=
\begin{bmatrix}
2 \\
2 \\
-4
\end{bmatrix}.
\]
By solving this system using the least-squares method, we can determine \( x_1 \) and \( x_2 \).
**Interactive Section:**
Please fill in the following to find the solution:
\[
x_1 = \_\_\_\_\_\_\_\_\_ ,
\]
\[
x_2 = \_\_\_\_\_\_\_\_\_ .
\]
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