system of linear equations
Enable an agent to recognise a system of linear equations, assess its solution status, and choose valid transformations or solution methods under explicit mathematical and numerical assumptions.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognise a system of linear equations, assess its solution status, and choose valid transformations or solution methods under explicit mathematical and numerical assumptions.
A system of linear equations is a collection of equations, each linear in a shared set of unknowns over a specified field, whose solutions are assignments that satisfy every equation simultaneously.
It can be Validate linearity relative to declared unknowns and construct Ax = b with a recoverable variable ordering.; Determine consistency and uniqueness exactly, numerically with stated tolerances, or conditionally on parameters.; Apply justified elimination or coordinate transformations while tracking their effects on solutions.; Construct an exact solution or describe all solutions using a particular solution and a null-space basis.; Choose a numerical method from verified matrix properties and check its output using residual and sensitivity evidence.; Identify redundant or contradictory equations and trace them to their originating constraints..
Distinguishing features
After simplification, every equation has the form sum(a_ij x_j) = b_i, with coefficients independent of the designated unknowns.
The equations constrain one shared assignment of unknowns simultaneously; solving each equation independently does not establish a system solution.
Products between unknowns, nonlinear powers, or nonlinear functions of unknowns place an equation outside scope unless an explicitly restricted equivalent linear formulation is established.
An augmented matrix represents a system only when its coefficient columns, right-hand-side column, unknown ordering, and scalar domain are identified.
Linearity depends on which symbols are unknowns: a x = b is linear in x when a is a fixed coefficient or parameter, but generally not linear jointly in a and x.
Scope
+ Equations, shared unknowns, coefficients, constants, and their representation as Ax = b
+ Scalar domain and any parameters on which coefficients or constants depend
+ Consistency, rank, degrees of freedom, and solution-set structure
+ Transformations that preserve solutions or explicitly relate different solution coordinates
+ Exact and numerical solution methods, residuals, and sensitivity
- Linear algebra as a discipline, including its institutions and literature
- Nonlinear systems and systems of inequalities
- Differential or difference equations whose unknowns are functions or sequences
- Optimisation objectives used to select approximate or preferred solutions
- The physical, economic, or other originating system and its independent validity
- Software libraries and computing infrastructure used to solve equations
Characteristics
- Scalar domain
- Named field, such as rational, real, complex, or finite field; other domains explicitly qualified Permitted operations and solution classifications depend on the domain; field-based rank criteria do not directly settle integer solvability.
- Equation and unknown counts
- Ordered pair (m, n) of nonnegative integer counts Determines matrix dimensions and the rectangular shape of the system without alone determining consistency or uniqueness.
- Homogeneity
- Homogeneous when b = 0; nonhomogeneous otherwise A homogeneous system always admits the zero solution and has a linear solution space over a field.
- Coefficient and augmented ranks
- Integer pair rank(A), rank([A|b]); numerical estimates include a tolerance Over a field, equal ranks establish consistency, and comparison with n determines uniqueness.
- Solution status
- Undetermined; inconsistent; unique solution; multiple solutions; conditional on parameters Controls whether to solve, parameterise, analyse parameter cases, or report infeasibility.
- Solution-space dimension
- n − rank(A) for a consistent system over a field; not applicable to an empty solution set Counts free directions; positive dimension implies infinitely many solutions over an infinite field but finitely many over a finite field.
- Coefficient structure
- Properties such as sparse, dense, banded, triangular, symmetric, Hermitian, or positive definite, where applicable Provides concrete eligibility conditions for specialised solution methods and storage choices.
- Residual
- Vector r = b − A x̂ and a named norm or scaled residual measure Measures equation satisfaction for a candidate solution without alone establishing closeness to an exact solution.
- Condition measure
- Named condition measure with norm, perturbation model, scaling, and applicability conditions Relates perturbations in data to solution sensitivity and helps interpret numerical accuracy.
- Originating constraints
- Links from equations and variables to source statements, measurements, or derivations Allows an agent to trace a disputed coefficient or equation without treating mathematical solvability as validation of the originating model.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 16 findings · 26 questions.
Linear formulation Establish what the equations mean and whether they form a linear system.
Linearity and valid algebraic operations cannot be assessed without distinguishing unknowns, parameters, and the scalar domain.
Unknowns and domain
Declare the symbols being solved for and the arithmetic governing them.
Declared linear dependence
Each equation must be linear in the designated unknowns, with coefficients independent of those unknowns.
- Which symbols are unknowns, which are fixed data or parameters, and in what scalar domain do solutions lie? definition
- After simplification, does every equation have coefficients independent of the unknowns, and are any domain restrictions required by that simplification? boundary
Matrix correspondence
Connect the written equations to coefficient and augmented matrices.
Recoverable matrix encoding
The representation Ax = b must preserve equation identities, unknown ordering, coefficients, and constants.
- What are A, x, and b, and which equation and unknown correspond to each row and column? definition
- Can every matrix entry be traced to the original equation or a documented derivation? provenance
Solution set Characterise whether simultaneous solutions exist and how much freedom they contain.
Equation counts alone cannot distinguish an inconsistent system from one with a unique solution or free variables.
Consistency and uniqueness
Establish solution status using domain-appropriate evidence.
Rank-based classification
Over a field, consistency requires rank(A) = rank([A|b]); a consistent system is unique precisely when rank(A) equals the number of unknowns.
- What are the coefficient and augmented ranks, and are they exact results or numerical estimates under a stated tolerance? measurement
- Do those ranks establish inconsistency, uniqueness, or multiple solutions under the declared scalar domain? definition
Solution parameterisation
Represent all solutions and any parameter-dependent changes in their structure.
Particular and null solutions
For a consistent system over a field, every solution is a particular solution plus an element of the null space of A.
- What particular solution and null-space basis describe all solutions, and how many free coordinates remain? definition
- For which coefficient-parameter values do consistency, rank, or the proposed parameterisation change? boundary
Equivalence and reduction Track transformations and the mathematical evidence exposed by elimination.
An agent must distinguish operations that preserve the original solution set from operations that change coordinates or discard constraints.
Valid transformations
Record the conditions under which transformed equations retain their intended meaning.
Solution-preserving operations
Invertible row operations preserve solutions over a field; transformations of unknown coordinates require an explicit mapping back.
- Which row operations or coordinate substitutions were applied, and how is their invertibility established? action
- Does any division require a parameter to be nonzero, and how is the zero case handled? boundary
Dependency and contradiction
Use reduction to expose repeated information and incompatible requirements.
Reduction certificates
Tracked equation combinations can certify redundancy or produce a contradiction of the form 0 = c with c nonzero.
- Which combinations of original equations demonstrate redundancy or inconsistency? provenance
- Which equations can be removed while preserving the complete solution set? action
Solution computation Select a suitable computational procedure and assess the reliability of its result.
Exact solvability, practical computation, and numerical accuracy are separate properties of a linear system.
Method eligibility
Match algorithm assumptions to verified coefficient structure and required output.
Structure-guided method
Dimensions, domain, sparsity, matrix properties, and desired exactness determine which elimination, factorisation, or iterative procedures are appropriate.
- What matrix properties and arithmetic requirements have been verified sufficiently to justify the selected method? measurement
- Is the required output one solution, all solutions, a consistency certificate, or a numerical approximation, and which procedure supplies it? action
Numerical reliability
Assess equation satisfaction and sensitivity under finite precision or uncertain data.
Residual and sensitivity evidence
Residuals measure equation satisfaction, while conditioning and error analysis determine what that evidence implies about solution accuracy.
- What residual or backward-error measure was computed, using which norm, scaling, precision, and acceptance threshold? measurement
- What sensitivity evidence supports the reported solution accuracy, particularly when the matrix is nearly rank deficient? measurement
Interpretation and use Connect algebraic results to their source meanings and permitted downstream uses.
A correct algebraic answer can still violate external restrictions or answer a modified problem without making that modification explicit.
Constraint meaning
Preserve variable meanings, units where relevant, and the origin of each equation.
Traceable formulation assumptions
Applied systems require traceable assumptions and compatible term units; abstract systems may have no physical units.
- What source statements, observations, or assumptions justify each equation and coefficient? provenance
- Where variables have units, are terms within each equation compatible, and what scaling converts computational coordinates back to those units? measurement
Downstream admissibility
Distinguish solving the equations from selecting or accepting an answer under additional requirements.
Additional restrictions and objectives
Nonnegativity, integrality, least-squares objectives, and regularisation introduce requirements beyond an unconstrained field-based system.
- Which additional restrictions determine whether an algebraic solution is admissible, and which neighbouring model owns those restrictions? boundary
- If no exact solution exists or several exist, what explicit objective authorises approximation or selection, and how will the resulting problem be identified? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- This describes a mathematical object, not a discipline, despite the batch context.
- The measurement statements assume finite systems; numerical conditioning statements concern real or complex coefficients.
- The listed kinds overlap: equation count alone does not establish consistency or uniqueness. Over a finite field, a nonunique solution set is finite.
- Which of these check these first hold for the sense of system of linear equations this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Homogeneous systems
- Nonhomogeneous systems
- Consistent systems
- Inconsistent systems
- Underdetermined systems
- Overdetermined systems
- Which of these kinds and varieties hold for the sense of system of linear equations this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Computing currents and voltages in linear electrical circuit models
- Solving discretized differential equations in engineering and physics
- Estimating parameters in linear least-squares problems
- Balancing chemical reaction equations through homogeneous linear constraints
- Solving linear input-output models in economics
- Which of these real-world use hold for the sense of system of linear equations this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Number of equations m and unknowns n - Positive integers; no universal typical range - count
- Rank of the coefficient matrix - 0 to min(m, n) - dimensionless integer
- Dimension of the solution set for a consistent system - n minus the rank of the coefficient matrix - dimension
- Condition number of a nonsingular square coefficient matrix in an induced matrix norm - At least 1, with no finite upper bound - dimensionless
- Which of these typical measurements hold for the sense of system of linear equations this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Inconsistent equations admit no simultaneous solution.
- A consistent system with coefficient rank below the number of unknowns does not determine a unique solution.
- Ill-conditioning can amplify small perturbations in coefficients or data into large solution changes.
- Floating-point roundoff and unsuitable elimination procedures can produce inaccurate numerical solutions.
- Treating a least-squares approximation as an exact solution can conceal incompatible constraints.
- Which of these failure modes and hazards hold for the sense of system of linear equations this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Linear algebra - Linear algebra is the broader mathematical field; a system of linear equations is an object studied within it.
- Linear equation - A linear equation supplies one constraint; a system requires simultaneous satisfaction of a collection of such constraints.
- Matrix - A matrix is an array of entries; a finite linear system can be represented as Ax = b using a coefficient matrix and a right-hand-side vector.
- System of nonlinear equations - A nonlinear system contains at least one equation that is not linear in the chosen unknowns.
- Linear least squares - Linear least squares minimizes a residual measure and can return an approximation when the corresponding linear system has no exact solution.
- Linear programming - Linear programming optimizes a linear objective subject to linear constraints; solving a linear system requires satisfying equalities without an optimization objective.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of system of linear equations this model covers, and on what evidence? provenance
What the second pass must settle
- Does an existing Vercy world model already cover this concept, requiring a registry link instead of a separate publication?
- Should this entry cover systems over general rings and integer-restricted systems directly, or reference specialised models while retaining a field-based core?
- Does the registry intend to include infinite systems or operator equations, beyond the finite scalar systems assumed in this draft?
- Which authoritative sources should support the published definitions, transformation rules, and numerical reliability criteria?
- What evidence and tolerance conventions should Vercy require when an agent reports numerical rank, approximate consistency, or solution accuracy?