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Give a matrix A and vector b so that Ax = b has no solution, but Ax = b has many least squares solutions. Give a basis for the least squares solutions. Give a matrix A and vector b so that Ax=b has no solution, and Ax=b has a unique least squares solutions. Give the least squares solution.

Give a matrix A and vector b so that Ax = b has no solution, but Ax = b has many least squares solutions. Give a basis for the least squares solutions. Give a matrix A and vector b so that Ax=b has no solution, and Ax=b has a unique least squares solutions. Give the least squares solution.

Advanced Engineering Mathematics
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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Please keep the matrices small and the entries 1’s or 0’s.

### Least Squares Solutions for Linear Systems #### Problem 1 Give a matrix \( A \) and vector \( \vec{b} \) so that \( A\vec{x} = \vec{b} \) has no solution, but \( A\vec{x} = \vec{b} \) has many least squares solutions. Give a basis for the least squares solutions. #### Problem 2 Give a matrix \( A \) and vector \( \vec{b} \) so that \( A\vec{x} = \vec{b} \) has no solution, and \( A\vec{x} = \vec{b} \) has a unique least squares solution. Give the least squares solution.
Transcribed Image Text:### Least Squares Solutions for Linear Systems #### Problem 1 Give a matrix \( A \) and vector \( \vec{b} \) so that \( A\vec{x} = \vec{b} \) has no solution, but \( A\vec{x} = \vec{b} \) has many least squares solutions. Give a basis for the least squares solutions. #### Problem 2 Give a matrix \( A \) and vector \( \vec{b} \) so that \( A\vec{x} = \vec{b} \) has no solution, and \( A\vec{x} = \vec{b} \) has a unique least squares solution. Give the least squares solution.
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