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3. Let A be an m x n matrix and be Rm. Given that Ax = b has two distinct solutions u and v. Explain why Ar= b has infinitely many solutions.

3. Let A be an m x n matrix and be Rm. Given that Ax = b has two distinct solutions u and v. Explain why Ar= b has infinitely many solutions.

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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Here is the transcribed text formatted for an educational website: --- ### Problem Statement: 3. Let \( A \) be an \( m \times n \) matrix and \( b \in \mathbb{R}^m \). Given that \( A\textbf{x} = b \) has two distinct solutions \( u \) and \( v \), explain why \( A\textbf{x} = b \) has infinitely many solutions. --- In this problem, we are given a linear system represented by a matrix equation \( A\textbf{x} = b \). Here, \( A \) is an \( m \times n \) matrix, and \( b \) is a vector in \( \mathbb{R}^m \). The problem statement reveals that there are at least two distinct solutions to this equation, \( u \) and \( v \). The objective is to explain why the existence of these two distinct solutions implies that there are infinitely many solutions to the equation \( A\textbf{x} = b \). --- Note: There are no graphs or diagrams associated with this text.
Transcribed Image Text:Here is the transcribed text formatted for an educational website: --- ### Problem Statement: 3. Let \( A \) be an \( m \times n \) matrix and \( b \in \mathbb{R}^m \). Given that \( A\textbf{x} = b \) has two distinct solutions \( u \) and \( v \), explain why \( A\textbf{x} = b \) has infinitely many solutions. --- In this problem, we are given a linear system represented by a matrix equation \( A\textbf{x} = b \). Here, \( A \) is an \( m \times n \) matrix, and \( b \) is a vector in \( \mathbb{R}^m \). The problem statement reveals that there are at least two distinct solutions to this equation, \( u \) and \( v \). The objective is to explain why the existence of these two distinct solutions implies that there are infinitely many solutions to the equation \( A\textbf{x} = b \). --- Note: There are no graphs or diagrams associated with this text.
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