Describe all least-squares solutions of the equation Ax = b. 1 0 1 1 10 1 A = 9 b = 110 110 1 The general least-squares solutions of Ax = b for the given matrix A and vector b are all vectors of the form x = +x3 with x3 free. (Simplify your answers.)
Describe all least-squares solutions of the equation Ax = b. 1 0 1 1 10 1 A = 9 b = 110 110 1 The general least-squares solutions of Ax = b for the given matrix A and vector b are all vectors of the form x = +x3 with x3 free. (Simplify your answers.)
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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![### Least-Squares Solutions of the Equation \( Ax = b \)
Given the matrix \( A \) and vector \( b \) as follows:
\[
A = \begin{pmatrix}
1 & 0 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
1 & 1 & 0
\end{pmatrix},
\quad
b = \begin{pmatrix}
1 \\
9 \\
5 \\
1
\end{pmatrix}
\]
The general least-squares solutions of \( Ax = b \) for the given matrix \( A \) and vector \( b \) are all vectors of the form:
\[
\hat{x} = \begin{pmatrix} \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \end{pmatrix} + x_3 \begin{pmatrix} \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \end{pmatrix}, \quad \text{with } x_3 \text{ free}.
\]
(Simplify your answers.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05b5783f-5507-4009-950e-cf9b7485abe1%2Faac79135-97b0-4d4d-8895-aef7792d4228%2Fuh9lza_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Least-Squares Solutions of the Equation \( Ax = b \)
Given the matrix \( A \) and vector \( b \) as follows:
\[
A = \begin{pmatrix}
1 & 0 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
1 & 1 & 0
\end{pmatrix},
\quad
b = \begin{pmatrix}
1 \\
9 \\
5 \\
1
\end{pmatrix}
\]
The general least-squares solutions of \( Ax = b \) for the given matrix \( A \) and vector \( b \) are all vectors of the form:
\[
\hat{x} = \begin{pmatrix} \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \end{pmatrix} + x_3 \begin{pmatrix} \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \\ \boxed{\phantom{0}} \end{pmatrix}, \quad \text{with } x_3 \text{ free}.
\]
(Simplify your answers.)
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