Find a least-squares solution of Ax = b by (a) constructing the normal equations for x and (b) solving for x. 1 - 3 3 b = 2 - 1 1 A = -2 a. Construct the normal equations for x without solving. (Simplify your answers.) LO 3.
Find a least-squares solution of Ax = b by (a) constructing the normal equations for x and (b) solving for x. 1 - 3 3 b = 2 - 1 1 A = -2 a. Construct the normal equations for x without solving. (Simplify your answers.) LO 3.
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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Then Solve for X
![### Finding a Least-Squares Solution of \( Ax = b \)
To find a least-squares solution of the equation \( Ax = b \), we proceed by:
**(a) Constructing the Normal Equations for \( \hat{x} \):**
Given matrices:
\[
A = \begin{bmatrix}
1 & -3 \\
-1 & 3 \\
0 & 2 \\
3 & 9
\end{bmatrix}, \quad
b = \begin{bmatrix}
3 \\
1 \\
-2 \\
5
\end{bmatrix}
\]
- Our goal is to create the normal equations for solving \( \hat{x} \).
\[
x = \begin{bmatrix}
\phantom{x} \\
\phantom{x}
\end{bmatrix}
\]
- **Instruction:** Simplify your answers, but do not solve at this step.
This process involves using the normal equation:
\[
A^TA\hat{x} = A^Tb
\]
The task here focuses on setting up this equation based on the given matrices. Stay tuned for the next steps, where we will proceed to solve for \( \hat{x} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6b254a-d25d-479c-b42b-752b8b5bbcbb%2F418eb8ea-57dd-4966-b806-2f05f2dc84f9%2F8eauj67_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding a Least-Squares Solution of \( Ax = b \)
To find a least-squares solution of the equation \( Ax = b \), we proceed by:
**(a) Constructing the Normal Equations for \( \hat{x} \):**
Given matrices:
\[
A = \begin{bmatrix}
1 & -3 \\
-1 & 3 \\
0 & 2 \\
3 & 9
\end{bmatrix}, \quad
b = \begin{bmatrix}
3 \\
1 \\
-2 \\
5
\end{bmatrix}
\]
- Our goal is to create the normal equations for solving \( \hat{x} \).
\[
x = \begin{bmatrix}
\phantom{x} \\
\phantom{x}
\end{bmatrix}
\]
- **Instruction:** Simplify your answers, but do not solve at this step.
This process involves using the normal equation:
\[
A^TA\hat{x} = A^Tb
\]
The task here focuses on setting up this equation based on the given matrices. Stay tuned for the next steps, where we will proceed to solve for \( \hat{x} \).
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