System of Linear Equations

Updated 01 Oct 2026

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What it is
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  a collection of equations linear in the unknowns, written $Ax=b$
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Solutions
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  consistent exactly when $\operatorname{rank}(A)=\operatorname{rank}([A\mid b])$ .
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  for $n$ unknowns, a consistent system has a unique solution when $\operatorname{rank}(A)=n$ ; otherwise it has free variables.
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  a square nonsingular matrix gives a unique solution for every right-hand side.
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Methods
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  direct: Gaussian Elimination, LU Decomposition, Cholesky Decomposition
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  iterative: Jacobi Method, Gauss-Seidel Method, Conjugate Gradient Method
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  inconsistent or overdetermined data can be handled through Unavailable reference
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Validation
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  residual: $r=b-A\hat x$ . interpret its size together with scaling and Condition Number

Linked references 3