Particular Solutions and Homogeneous Solutions

Q: If two solvers give different answers, is one of them wrong?

Not necessarily. Subtract the answers and apply $A$. If the result is zero to rounding, both are correct and differ only by a null space element. Only a significant residual indicates an actual error.

Overview

Two questions about a linear system look independent: does a solution exist, and how many are there? The particular-plus-homogeneous decomposition shows they are not. Existence is a property of the pair $(A, \vec{b})$ ; multiplicity is a property of $A$ alone. Once a single solution has been found by any means, the entire solution set is that solution translated by the null space of the coefficient matrix.

The statement is short. If $\vec{w}$ is one solution of $A\vec{x} = \vec{b}$ , then $\vec{y}$ is a solution if and only if $\vec{y} = \vec{w} + \vec{z}$ for some $\vec{z}$ with $A\vec{z} = \vec{0}$ . Both directions are quick consequences of expressing solutions as linear combinations of the columns: adding a homogeneous solution adds the zero vector to the combination, and subtracting two solutions cancels the constant vector.

This is the linear-algebra form of superposition, and engineers meet it long before they meet the theorem. A statically indeterminate structure carries a particular equilibrium force field plus any self-stress state; a driven oscillator responds with a particular forced response plus any free vibration; a controlled manipulator executes a particular joint motion plus any self-motion. In each case the particular part answers the demand and the homogeneous part is the residual freedom of the system.

The practical corollaries are sharp. A singular coefficient matrix has a non-trivial null space, so a system built on it has either no solutions or infinitely many — never exactly one. Any two solutions differ by a null space element, which gives a cheap consistency check on independently computed answers. And the null space needs to be computed only once, however many right-hand sides are later applied.

Definition

Particular Solution

Any single vector $\vec{w} \in \mathbb{C}^n$ satisfying $A\vec{w} = \vec{b}$ . A consistent system has at least one; it has exactly one only when $\mathcal{N}(A) = \{\vec{0}\}$ . No particular solution is privileged — a solver's output, the fixed vector of a vector-form description and a minimum-norm least-squares answer are all equally valid choices.

The term carries no implication of being special or minimal. It means only that one specific solution has been singled out for use as a reference point.

Homogeneous Solution

A vector $\vec{z} \in \mathbb{C}^n$ satisfying $A\vec{z} = \vec{0}$ ; that is, an element of the null space $\mathcal{N}(A)$ . The set of all such vectors is closed under addition and scalar multiplication, so it is a subspace, and it always contains $\vec{0}$ .

Particular Solution Plus Homogeneous Solutions

PSPHS

Let $\vec{w}$ be one solution of the system with coefficient matrix $A$ and constant vector $\vec{b}$ . Then a vector $\vec{y}$ is a solution of that system if and only if $\vec{y} = \vec{w} + \vec{z}$ for some $\vec{z} \in \mathcal{N}(A)$ . Equivalently, the solution set is the translate $\mathcal{S} = \vec{w} + \mathcal{N}(A)$ .

Concepts

Why the decomposition holds

Read $A\vec{x} = \vec{b}$ as the statement that $\vec{b}$ is the linear combination of the columns of $A$ with coefficients taken from $\vec{x}$ . If $A\vec{w} = \vec{b}$ and $A\vec{z} = \vec{0}$ , then the combination with coefficients $\vec{w} + \vec{z}$ is the sum of the two combinations, which is $\vec{b} + \vec{0} = \vec{b}$ ; so $\vec{w} + \vec{z}$ is a solution.

Conversely, if $\vec{y}$ and $\vec{w}$ are both solutions, the combination with coefficients $\vec{y} - \vec{w}$ equals $\vec{b} - \vec{b} = \vec{0}$ , so $\vec{y} - \vec{w}$ lies in the null space and $\vec{y} = \vec{w} + (\vec{y} - \vec{w})$ has the required form. Both directions rest only on distributivity of the column combination over vector addition.

Solution sets are affine, not linear

Unless $\vec{b} = \vec{0}$ , the solution set does not contain the origin and is not closed under addition: if $A\vec{y}_1 = A\vec{y}_2 = \vec{b}$ , then $A(\vec{y}_1 + \vec{y}_2) = 2\vec{b}$ . What is closed is the set of differences. A solution set of a consistent inhomogeneous system is therefore a coset — a rigid translate of a subspace — and its dimension as a geometric object is the nullity $n - r$ .

This distinction matters when a solution set is passed to code that assumes vector space structure. Averaging two solutions is legitimate (the coefficients sum to one); adding them is not.

Singular means never unique

If the square matrix $A$ is singular, its null space contains a non-zero vector $\vec{z}$ , and then $\vec{z}$ can be scaled arbitrarily. So if the system has even one solution $\vec{w}$ , it has the infinite family $\vec{w} + \alpha\vec{z}$ . Hence for a singular coefficient matrix a system is either inconsistent or has infinitely many solutions.

Read the other way, this is the negation of the nonsingular case: a nonsingular matrix yields a unique solution for every right-hand side, and only a nonsingular matrix does. The two statements together classify square systems completely.

The difference test

The corollary that $\vec{y}_1 - \vec{y}_2 \in \mathcal{N}(A)$ for any two solutions is more useful than it looks. It gives an independent check on solvers: compute a solution by two different routes, subtract, and confirm that the difference is annihilated by $A$ . A non-zero residual $A(\vec{y}_1 - \vec{y}_2)$ signals an arithmetic error rather than an acceptable difference of parametrisation.

It also explains why two correct programs can return visibly different answers to the same under-determined system without either being wrong: they have chosen different points of the same coset.

Reuse across right-hand sides

The null space depends only on $A$ . In a design study where the geometry is fixed and the loading varies, $\mathcal{N}(A)$ is computed once and each new load case requires only one particular solution. This is the structural argument behind reusing a factorisation across right-hand sides, and it is why the expensive part of a solve is charged to the matrix rather than to the load.

Conversely, changing $A$ invalidates everything: a modified geometry or a new member changes both the particular solution and the space of freedoms.

The same structure elsewhere

The decomposition is not special to matrices. For any linear map $T$ , the pre-image of a vector $\vec{v}$ under $T$ is either empty or a translate of the kernel $\mathcal{K}(T)$ . For a linear differential operator the same statement is the familiar rule that the general solution is a particular integral plus the complementary function.

Recognising the pattern early saves effort later: once a problem is known to be linear, the shape of its solution set is settled before any computation begins.

Describing a solution set from one solution

Obtain one solutionFind any

\vec{w}

with

A\vec{w} = \vec{b}

— by elimination, by a factorisation, or by inspection of the physics.

Confirm it really is a solutionSubstitute into the original system. Everything that follows depends on

\vec{w}

being exact, not approximate.

Compute the null spaceSolve the homogeneous system

A\vec{z} = \vec{0}

and record a basis

\vec{z}_1, \ldots, \vec{z}_{n-r}

Assemble the solution setWrite

\mathcal{S} = \{\, \vec{w} + \alpha_1\vec{z}_1 + \cdots + \alpha_{n-r}\vec{z}_{n-r} \,\}

over all scalar choices.

Interpret the two partsThe particular part answers the demand; the homogeneous part is the residual design freedom, to be spent on a secondary objective.

Equations

Particular plus homogeneous

EQ-PSHS-01

A \vec{w} = \vec{b}, A \vec{z} = \vec{0} \Rightarrow A (\vec{w} + \vec{z}) = \vec{b}

Adding any homogeneous solution to a particular solution produces another solution. This is the easy direction of the theorem.

Difference of two solutions

EQ-PSHS-02

A {\vec{y}}_{1} = A {\vec{y}}_{2} = \vec{b} \Rightarrow A ({\vec{y}}_{1} - {\vec{y}}_{2}) = \vec{0} \Rightarrow {\vec{y}}_{1} - {\vec{y}}_{2} \in N (A)

The converse direction, and a practical cross-check between independently computed solutions.

Solution set as a translate

EQ-PSHS-03

S = \vec{w} + N (A) = {\vec{w} + \vec{z} ∣ \vec{z} \in N (A)}

The complete description. The set is a coset of the null space; its geometric dimension is $n - r$ .

Explicit parametrisation

EQ-PSHS-04

S = {\vec{w} + α_{1} {\vec{z}}_{1} + α_{2} {\vec{z}}_{2} + \dots + α_{n - r} {\vec{z}}_{n - r} ∣ α_{i} \in ℂ}

Written over a basis $\{\vec{z}_1, \ldots, \vec{z}_{n-r}\}$ of the null space. Each choice of the $\alpha_i$ gives exactly one solution.

Trichotomy for square systems

EQ-PSHS-05

N (A) \neq {\vec{0}} \Rightarrow | S | \in {0, \infty}

A singular coefficient matrix admits no unique solution: the system is either inconsistent or has infinitely many solutions.

Uniqueness criterion

EQ-PSHS-06

| S | = 1 \Leftrightarrow S \neq \emptyset and N (A) = {\vec{0}}

Existence and uniqueness are genuinely independent conditions: the first depends on the right-hand side, the second only on the matrix.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Coefficient matrix	The $m \times n$ matrix of the system	m x n matrix
$\vec{b}$	Constant vector	The right-hand side; the demand placed on the system	vector in C^m
$\vec{w}$	Particular solution	One chosen vector satisfying $A\vec{w} = \vec{b}$	vector in C^n
$\vec{z}$	Homogeneous solution	A vector satisfying $A\vec{z} = \vec{0}$	element of N(A)
$\vec{y}$	General solution	An arbitrary member of the solution set	vector in C^n
$N (A)$	Null space	The set of all homogeneous solutions; a subspace	subspace of C^n
$S$	Solution set	All vectors satisfying the inhomogeneous system	subset of C^n
$n - r$	Nullity	Dimension of the null space; the number of independent freedoms	non-negative integer
$α_{i}$	Free parameter	Coefficient selecting a particular homogeneous contribution	complex scalar

Worked Numerical Example

Problem statement

A four-branch supply network must meet three demand balances. One feasible dispatch is known from operating experience. Show that it and a second dispatch found by elimination differ by an element of the null space, and describe every feasible dispatch.

State the system
The balances are $x_1 + x_2 + x_3 + x_4 = 4$ , $x_1 + 2x_2 + 3x_3 + 4x_4 = 10$ and $2x_1 + 3x_2 + 4x_3 + 5x_4 = 14$ . The third is the sum of the first two, so one equation is redundant — a common artefact of writing down more balances than are independent.
$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 5 \end{matrix}], \vec{b} = [\begin{matrix} 4 \\ 10 \\ 14 \end{matrix}]$
Check the known dispatch
Operating experience offers $\vec{w} = (1,\, 1,\, 1,\, 1)$ . Substituting gives $1+1+1+1 = 4$ , $1+2+3+4 = 10$ and $2+3+4+5 = 14$ . All three balances hold, so $\vec{w}$ is a genuine particular solution.
Row-reduce the augmented matrix
Elimination gives the reduced form below. Column $5$ is not a pivot column, confirming consistency; $r = 2$ , $D = \{1, 2\}$ and $F = \{3, 4\}$ , so the nullity is $n - r = 4 - 2 = 2$ .
$[\begin{matrix} 1 & 0 & - 1 & - 2 & - 2 \\ 0 & 1 & 2 & 3 & 6 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$
Extract a basis for the null space
Rows $1$ and $2$ give $x_1 = x_3 + 2x_4$ and $x_2 = -2x_3 - 3x_4$ for the homogeneous system. Taking $(x_3, x_4) = (1, 0)$ and then $(0, 1)$ yields two independent homogeneous solutions.
${\vec{z}}_{1} = [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], {\vec{z}}_{2} = [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]$
Verify the null space vectors
For $\vec{z}_1$ : $1 - 2 + 1 + 0 = 0$ , $1 - 4 + 3 + 0 = 0$ and $2 - 6 + 4 + 0 = 0$ . For $\vec{z}_2$ : $2 - 3 + 0 + 1 = 0$ , $2 - 6 + 0 + 4 = 0$ and $4 - 9 + 0 + 5 = 0$ . Both are annihilated by $A$ .
Compare with a second dispatch
Elimination with both free variables set to zero returns $\vec{y} = (-2,\, 6,\, 0,\, 0)$ , which also satisfies the balances: $-2 + 6 = 4$ , $-2 + 12 = 10$ , $-4 + 18 = 14$ . The two dispatches look nothing alike, yet their difference must lie in the null space.
$\vec{y} - \vec{w} = [\begin{matrix} - 2 \\ 6 \\ 0 \\ 0 \end{matrix}] - [\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} - 3 \\ 5 \\ - 1 \\ - 1 \end{matrix}] = (- 1) {\vec{z}}_{1} + (- 1) {\vec{z}}_{2}$
Write the full solution set
With one particular solution and a null space basis, every feasible dispatch is now described exactly.
$S = {[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}] + α_{1} [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}] + α_{2} [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}] ∣ α_{1}, α_{2} \in ℂ}$

Result

The two dispatches differ by $-\vec{z}_1 - \vec{z}_2$ , exactly as the theorem requires, so both are members of the same two-parameter family. Operationally, the demand fixes two degrees of freedom and leaves two free: those two directions are circulating patterns that move supply between branches without changing what is delivered, and they are the budget available for a secondary objective such as minimising cost or losses.

Applications & Industry Use

Structural engineering

Equilibrium plus self-stress

In an indeterminate frame, any force distribution in equilibrium with the applied load is a particular solution, and the null space of the equilibrium matrix is the space of self-stress states. Prestressing and cable tensioning are the deliberate selection of a null space element to improve the response without disturbing equilibrium.

Power systems

Economic dispatch on a redundant network

Generation schedules meeting demand form an affine set; the null space describes redistributions that leave every bus balance satisfied. Dispatch optimisation searches exactly this set, so characterising it first converts an equality-constrained problem into an unconstrained one over the free parameters.

Robotics

Self-motion of redundant manipulators

Joint velocities producing a commanded end-effector twist are a particular solution plus any null space velocity. Null-space projection uses that freedom to avoid joint limits, singularities and obstacles while the tool follows its path unchanged.

Mechanical vibration

Forced response plus free response

The response of a linear structure to harmonic excitation is a particular forced solution superposed on free vibration determined by the initial conditions. The decomposition is what allows steady-state and transient behaviour to be designed for separately.

Chemical engineering

Reaction extents and mass balance closure

A measured composition change satisfying elemental balances is a particular solution; the null space of the atomic matrix gives the independent reactions. Data reconciliation exploits this to adjust measurements only along directions that keep the balances closed.

Statistics & modelling

Non-identifiable parameters

In a rank-deficient regression, the fitted parameters are a particular solution plus any vector in the design matrix null space. Contrasts orthogonal to that null space are estimable; contrasts inside it are not, which determines what may honestly be reported.

Design Considerations

Verify the particular solution before building on it

The whole description hinges on $\vec{w}$ being an exact solution. If it came from a floating-point solver, check the residual $\|A\vec{w} - \vec{b}\|$ relative to $\|\vec{b}\|$ before declaring the set. An approximate particular solution shifts the entire family.

Choose the particular solution to mean something

Since any solution serves, choose one with engineering content: a minimum-norm dispatch, a zero-prestress force state, a nominal operating point. Choosing the accidental output of an elimination is legitimate but hands an arbitrary variable ordering a role it should not have.

Compute the null space once per matrix

The homogeneous part depends only on $A$ . In parameter sweeps over the right-hand side, hoist that computation out of the loop; in sweeps over the matrix, do not, because a small change in $A$ can change the nullity discontinuously.

Do not add two solutions

Solution sets of inhomogeneous systems are not closed under addition. Convex or affine combinations with coefficients summing to one remain solutions; arbitrary sums do not. Code operating on solution sets must respect that distinction explicitly.

Treat a large nullity as a modelling signal

A high-dimensional null space usually means the model is under-specified rather than that the design is unusually flexible: missing constraints, duplicated equations or an over-parametrised description. Inspect the null space basis for physical meaning before spending the freedom.

Numerical nullity is a tolerance decision

In floating point the null space is characterised by small rather than zero singular values, and the boundary is a choice. State the tolerance, prefer an SVD-based null space to a row-reduced one for measured data, and report the smallest retained singular value alongside the nullity.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Fixes the notation for sets, translates and the null space symbol $\mathcal{N}(A)$ used to state the decomposition.
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Minimum-norm particular solutions to rank-deficient systems come from `xGELSD`; an orthonormal null space basis is obtained from the trailing right singular vectors returned by `xGESVD`.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Governs the residual arithmetic used to certify a particular solution and to test whether the difference of two solutions is genuinely annihilated by the matrix.
`ISO 2394`	General principles on reliability for structures	Underpins the equilibrium and superposition arguments used in structural analysis, where the particular-plus-self-stress decomposition is the standard treatment of indeterminate systems.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the set-builder and vector expressions on this page so that the affine structure remains machine-readable.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact rational arithmetic	Small integer systems where the null space basis is to be reported symbolically or checked for exact membership.	Gives an unambiguous nullity with no tolerance, but scales badly and can produce unwieldy fractions in the basis vectors.
IEEE 754 binary64 with SVD-based null space	Measured or simulated data where the matrix is only approximately rank-deficient.	Robust and gives an orthonormal basis with a numerical rank criterion, at roughly ten times the cost of elimination.
IEEE 754 binary64 with row reduction	Well-conditioned systems where speed matters and the pivot pattern is trustworthy.	Cheap and gives basis vectors with a readable 0/1 pattern, but the rank decision is fragile near dependence.
QR with column pivoting	Moderate-size dense problems needing both a particular solution and a rank estimate in one factorisation.	Cheaper than SVD and rank-revealing in practice, but can fail on contrived matrices and gives a less stable basis.
Sparse iterative solve with null space deflation	Very large network or finite element systems with a small, known null space.	Keeps memory tractable and converges well once the null space is deflated, but requires the null space to be characterised in advance, usually from the physics.
Complex scalars	Phasor and frequency-domain systems where the right-hand side and null space are complex.	The decomposition is unchanged and applies verbatim, at double storage and roughly four times the multiply cost.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Cost profile

Obtaining one particular solution costs a factorisation, roughly $\tfrac{2}{3}n^3$ for a dense square system, plus $O(n^2)$ per additional right-hand side. Obtaining the null space costs a rank-revealing factorisation, comparable for elimination and around an order of magnitude more for an SVD. Once both are in hand, generating any number of further solutions is $O(n(n-r))$ each.

Certifying the decomposition

Two residual checks certify everything: $\|A\vec{w} - \vec{b}\|$ small relative to $\|\vec{b}\|$ , and $\|A\vec{z}_j\|$ small relative to $\|A\| \|\vec{z}_j\|$ for each basis vector. Checking individual members of the family adds nothing, because any member is a combination of quantities already certified.

Library behaviour

SymPy's linsolve returns a parametrised family that is precisely a particular solution plus a null space combination. NumPy's lstsq returns the minimum-norm particular solution and discards the null space; scipy.linalg.null_space supplies it separately with an explicit rcond. MATLAB's backslash on a rank-deficient system returns a basic solution with zeros in non-pivot positions, which is a different particular solution again — not an error.

Reconciling two solvers

When two tools return different answers to the same under-determined system, subtract them and apply the matrix. If the product is at rounding level the tools agree, having picked different points of the same coset. Only a significant residual indicates a real discrepancy. This test should be built into any regression suite covering under-determined solves.

Exploiting the freedom

To optimise a secondary objective within the solution set, parametrise as $\vec{x} = \vec{w} + U\vec{t}$ with $U$ holding the null space basis, and optimise over $\vec{t}$ unconstrained. This eliminates the equality constraints entirely and is the standard null-space method for equality-constrained quadratic programming.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Applying the theorem to an inconsistent system	high	Assuming a particular solution exists without checking consistency.	The hypothesis is that at least one solution exists. Test consistency first; if it fails, the solution set is empty and the null space is irrelevant.
Using an approximate particular solution	high	Taking a floating-point solve at face value when the matrix is ill-conditioned.	Compute and report the relative residual before using the vector as the anchor of the whole family.
Adding two solutions	high	Treating the solution set as if it were a subspace.	Only differences of solutions, and affine combinations with coefficients summing to one, stay in the set. Add homogeneous solutions to a particular one, never solutions to each other.
Expecting a unique solution from a singular matrix	medium	Assuming that a square system always determines its unknowns.	Check the null space. A non-trivial null space forces the count of solutions to be zero or infinite; a solver that returns one answer has silently chosen among many.
Recomputing the null space per right-hand side	low	Placing the homogeneous solve inside the load-case loop.	The null space depends only on the matrix. Compute it once and reuse it for every load case that shares the same geometry.
Confusing nullity with the number of zero rows	medium	Reading the count of freedoms off the row count of the reduced form.	Nullity is $n - r$ , computed from the number of unknowns. Zero rows count redundant equations, which is a different quantity unless $m = n$ .
Reporting a basis-dependent answer as canonical	low	Treating a particular null space basis as though it were determined by the problem.	Any basis describes the same subspace. When exchanging results, compare the spans, or agree on an orthonormal basis with a fixed sign convention.
Spending the null space freedom without checking physical validity	medium	Adding a null space element that satisfies the balances but violates an unmodelled constraint such as non-negativity or a capacity limit.	Validate every generated member against the constraints that were left out of the linear model.

FAQs

Does it matter which particular solution I choose?

Not for the description of the set: $\vec{w} + \mathcal{N}(A)$ is the same set for every solution $\vec{w}$ . It matters for readability and for reporting, since a minimum-norm or physically nominal choice communicates more than an arbitrary one produced by elimination.

Why can a singular system never have exactly one solution?

Because a singular matrix has a non-zero null space vector $\vec{z}$ . If one solution $\vec{w}$ exists then $\vec{w} + \alpha\vec{z}$ is a solution for every scalar $\alpha$ , giving infinitely many. So the only possibilities are none or infinitely many.

How is this different from the vector form of a solution set?

It is the same fact at a different level of abstraction. The vector form is the constructive recipe that reads a specific particular solution and a specific null space basis off the reduced row-echelon form; this theorem states the structure without reference to any construction, and therefore applies to solutions found by any method.

Can I use the decomposition when the system is over-determined?

Yes, provided it is consistent. Consistency, not the shape of the matrix, is the hypothesis. If an over-determined system is inconsistent, the analogous statement applies to the least-squares problem: the set of minimisers is one minimiser plus $\mathcal{N}(A)$ .

What does the null space mean physically?

It is the set of changes that the system cannot see: internal redistributions producing no observable effect on the right-hand side. Self-stress states, circulating flows, manipulator self-motions and non-identifiable parameter directions are all instances.

If two solvers give different answers, is one of them wrong?

Not necessarily. Subtract the answers and apply $A$ . If the result is zero to rounding, both are correct and differ only by a null space element. Only a significant residual indicates an actual error.

Does the same structure hold for linear differential equations?

Yes, and for any linear operator. The general solution is a particular integral plus the complementary function, which is the kernel of the differential operator. The algebra is identical; only the space in which the vectors live has changed.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section LC, Subsection PSHS. Licensed under the GNU Free Documentation License v1.2.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013.
Nocedal, J. and Wright, S. J. Numerical Optimization, 2nd edition. Springer, 2006. (Null-space methods for equality-constrained problems.)
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.

AI Suggested Questions

Given a rank-deficient system, produce two visibly different particular solutions and verify that their difference lies in the null space.
Show how the null-space method converts an equality-constrained quadratic programme into an unconstrained one.
Why is the set of solutions to a consistent inhomogeneous system an affine set rather than a subspace, and what operations does it support?
Compare the particular solution returned by MATLAB backslash, NumPy lstsq and SymPy linsolve on the same rank-deficient system.
Explain how the particular-plus-homogeneous structure appears in the general solution of a linear ordinary differential equation.
How should the numerical nullity be reported so that a downstream engineer can judge whether a freedom is genuine?

Related Calculators

Null Space Basis Calculator

Compute a basis for the null space of a matrix, with a choice of pivot-based or orthonormal output.

Solution Set Explorer

Enter a particular solution and a null space basis and generate, verify and visualise members of the solution family.

Residual Checker

Evaluate $\|A\vec{x} - \vec{b}\|$ relative to $\|\vec{b}\|$ to certify a claimed particular or homogeneous solution.