Difference between consistent and inconsistent systems in least squares context

Question

Hey there! 👋 Ever get confused about consistent vs. inconsistent systems in least squares? It's a common head-scratcher in linear algebra. Let's break it down simply! 🧮

michele.watson · Accepted Answer

📚 Consistent Systems in Least Squares
In the context of least squares, a consistent system is one where there exists at least one solution to the equation $Ax = b$. This means that the vector $b$ lies within the column space of the matrix $A$.

📈 Inconsistent Systems in Least Squares
An inconsistent system in least squares arises when there is no exact solution to the equation $Ax = b$. This happens when the vector $b$ does not lie within the column space of the matrix $A$. Because there’s no solution, we seek the 'best' approximate solution, which minimizes the error. This best approximate solution is called the least squares solution.

📊 Consistent vs. Inconsistent Systems: A Comparison

Feature
 Consistent System
 Inconsistent System

Definition
 $Ax = b$ has at least one solution.
 $Ax = b$ has no solution.

Location of $b$
 $b$ is in the column space of $A$.
 $b$ is not in the column space of $A$.

Error
 The error, $||Ax - b||$, can be zero.
 The error, $||Ax - b||$, cannot be zero; we find the least squares solution instead.

Least Squares Solution
 The least squares solution is an exact solution.
 The least squares solution provides the 'best' approximate solution that minimizes $||Ax - b||$.

Normal Equation
 Solutions to $A^TAx = A^Tb$ provide the solution to $Ax=b$.
 Solutions to $A^TAx = A^Tb$ give the least squares solution to $Ax=b$.

key Takeaways

🔍 Consistency: A consistent system admits an exact solution, implying the vector $b$ lies within the column space of matrix $A$.
  💡 Inconsistency: An inconsistent system lacks an exact solution. This necessitates finding a 'best-fit' solution, known as the least squares solution.
  📝 Least Squares: Least squares provides the tools to find the closest approximation when the system is inconsistent, effectively minimizing the error between $Ax$ and $b$.
  ➗ Normal Equations: The normal equation $A^TAx = A^Tb$ are used to find solutions to $Ax=b$ in consistent systems and least squares solutions in inconsistent systems.
  📐 Column Space: Understanding the column space of a matrix is critical in identifying whether a system is consistent or inconsistent.

Difference between consistent and inconsistent systems in least squares context

1 Answers

📚 Consistent Systems in Least Squares

📈 Inconsistent Systems in Least Squares

📊 Consistent vs. Inconsistent Systems: A Comparison

key Takeaways

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Feature	Consistent System	Inconsistent System
Definition	$Ax = b$ has at least one solution.	$Ax = b$ has no solution.
Location of $b$	$b$ is in the column space of $A$.	$b$ is not in the column space of $A$.
Error	The error, $\|\|Ax - b\|\|$, can be zero.	The error, $\|\|Ax - b\|\|$, cannot be zero; we find the least squares solution instead.
Least Squares Solution	The least squares solution is an exact solution.	The least squares solution provides the 'best' approximate solution that minimizes $\|\|Ax - b\|\|$.
Normal Equation	Solutions to $A^TAx = A^Tb$ provide the solution to $Ax=b$.	Solutions to $A^TAx = A^Tb$ give the least squares solution to $Ax=b$.