2 0 -1 1 -2 2 A = 2 -1 0 1 -1 6. ||
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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![On this educational page, we address the problem of finding the least squares solution for the equation \( A\vec{x} = \vec{b} \).
The matrix \( A \) and the vector \( \vec{b} \) are given as:
\[
A = \begin{bmatrix}
2 & 0 & -1 \\
1 & -2 & 2 \\
2 & -1 & 0 \\
0 & 1 & -1
\end{bmatrix}
\]
\[
\vec{b} = \begin{bmatrix}
0 \\
6 \\
0 \\
6
\end{bmatrix}
\]
(You can use a computer for all the matrix multiplications!)
For further clarity, in finding the least squares solution to \( A\vec{x} = \vec{b} \), we are determining the vector \(\vec{x}\) that minimizes the expression \(\|A\vec{x} - \vec{b}\|^2\). This is particularly useful when the system of equations represented by \( A\vec{x} = \vec{b} \) does not have an exact solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2520f39-8b3c-4b2b-be15-831ac0e5241f%2F7c621ec6-beee-4ecb-b0cb-c9db754e4860%2Fj5qwu4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:On this educational page, we address the problem of finding the least squares solution for the equation \( A\vec{x} = \vec{b} \).
The matrix \( A \) and the vector \( \vec{b} \) are given as:
\[
A = \begin{bmatrix}
2 & 0 & -1 \\
1 & -2 & 2 \\
2 & -1 & 0 \\
0 & 1 & -1
\end{bmatrix}
\]
\[
\vec{b} = \begin{bmatrix}
0 \\
6 \\
0 \\
6
\end{bmatrix}
\]
(You can use a computer for all the matrix multiplications!)
For further clarity, in finding the least squares solution to \( A\vec{x} = \vec{b} \), we are determining the vector \(\vec{x}\) that minimizes the expression \(\|A\vec{x} - \vec{b}\|^2\). This is particularly useful when the system of equations represented by \( A\vec{x} = \vec{b} \) does not have an exact solution.
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