HomeCalculatorsMathLinear AlgebraLeft Nullspace Calculator
Math calculator

Left Nullspace Calculator

Left-nullspace (cokernel) of a small real square matrix as ker(Aᵀ). Same engine as Linear System. Trivial left kernel is {0}. Not Jordan form. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
A basis for ker Aᵀ of a small real square matrix (order 2–4), with dim = n − rank A.
Scope
Real square matrices
Verification
Engine tested · Source checked · v1.14.16 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.16 · CVP protocol 1.0.0-proposed · Evidence 2026-10-30.cholesky-numeric-contract-v2
Verification revision
2026-10-30.cholesky-numeric-contract-v2 · 10/10 property · digest 2f8af10fde34
Legacy regression
58/58 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 24/24 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
478/478 golden · 8/8 CVP boundary · 78/78 invalid · 10/10 property · 2/2 metamorphic · 3/3 round-trip · 67/67 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
Sources
Sources
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 478/478 oracle-backed golden · 78/78 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.16 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

Dimensiondim ker Aᵀ = n − rank A
Transposeleft-null A = ker Aᵀ
iSquare matrices only, order ≤ 4. Reuses Gauss–Jordan already used by inverse, rank, RREF, and nullspace, applied to Aᵀ. Not a CAS. Not Jordan form. Float64. Pivot tests are scale-aware (ε·max(m,n)·‖A‖_max).

How to use

1

Enter a square matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix. Nullspace, column space, and row space are sibling discovery URLs.

2

Read the basis

A full-rank matrix returns {0}. Otherwise each free column of RREF(Aᵀ) yields a basis vector y with residual ‖yᵀ A‖.

Example calculations

Common configurations with formula and result.

ϟ

Rank 1 (not symmetric)

[[1,2],[3,6]]

dim = 1
[−3, 1]
ϟ

Identity

I₂

rank = n
{0}

Left Nullspace calculator specification

Version 1.14.16 · Engine tested

Calculation status

Review policy · Evidence

Definition
ker Aᵀ = { y | yᵀ A = 0 }. This page is mode=leftnullspace on math.linear_algebra. A basis is the right kernel of Aᵀ from RREF. Not a second engine. Not Jordan form.
What it calculates
A basis for ker Aᵀ of a small real square matrix (order 2–4), with dim = n − rank A.
Inputs
  • order
  • a11…
Outputs
  • basis
  • dimension
  • rank
  • trivial
  • residual_max
Formula
dim ker Aᵀ = n − rank A
Assumptions
  • Real square matrices
  • Order ≤ 4
  • Planar ℝ² figure when order=2; order 3+ omits it
  • Float64
  • Not a CAS
  • Not Jordan form as a product
Units
  • dimensionless
Boundary conditions
  • full rank → trivial left-nullspace {0}
  • unknown mode (including jordan-form) → INVALID_MODE
Example
[[1,2],[3,6]] → basis [[-3,1]], dim=1
Validation cases

2 published on this page · 58/58 tests · Production surface contract 6/6 · View evidence

  • order=2 a11=1 a12=2 a21=3 a22=6 → dimension=1, basis=[[-3,1]]
  • mode=jordan-form order=2 a11=1 a12=0 a21=0 a22=1 → INVALID_MODE
Sources
Calculation version
1.14.16

Background

Interpretation and common distinctions.

What does this left nullspace calculator compute?

ker Aᵀ = { y | yᵀ A = 0 }.

The published formula or method is dim ker Aᵀ = n − rank A. The following references support the stated mathematics or method:

Which inputs and limits should you check?

Rank, inverse, subspace, and least-squares decisions depend on the stated tolerance and conditioning. Inspect residuals and warnings before interpreting a nearly singular matrix.

The worked example uses [[1,2],[3,6]] with dim = 1. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.linear_algebra v1.14.16 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Input replay and numerical range

All entries share math.linear_algebra v1.14.16. Enter every cell as a finite decimal, including scientific notation. Empty cells and nonzero decimals below the binary64 range are rejected. QR, SVD, and least squares accept dimensions from 2 to 4; other operations require square matrices. Matching aliases may be supplied, while conflicting aliases or nested and flat cells are rejected.

A singular system may be consistent with infinitely many solutions or inconsistent; read the status and warnings. Tolerance-based rank and subspace decisions are not exact symbolic certificates. Results outside the numerical range refuse with NUMERIC_PRECISION_LIMIT. An unavailable auxiliary determinant is omitted with a warning, rather than reported as zero or null.

Calculations run locally in the browser. Shared query links contain inputs, are sent to the server when opened, and may appear in server or third-party logs. Anyone receiving a shared link can read its inputs. JSON and CSV exports include the original inputs, complete structured result, warnings, and calculation version.

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Condition Number Calculator, Least Squares Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator . Explore all Linear Algebra.

→ Cholesky Factorization Calculator Lower Cholesky factorization of a small real symmetric positive-definite matrix. Same engine as Linear System. Not LU. Not a CAS. Runs locally. › → Column Space Calculator Column space (image) of a small real square matrix from RREF pivot columns. Same engine as Linear System. Trivial image is {0}. Left-nullspace is a sibling. Runs locally. › → Condition Number Calculator κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally. › → Least Squares Calculator Linear least squares for a small real matrix, square or rectangular. Full column rank uses QR; otherwise SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, Cholesky, QR, SVD, least squares, RREF, condition number, nullspace, column space, rowspace, left-nullspace, and eigenvalues. Same engine as the matrix workspace. Not a CAS. Runs locally. › → LU Factorization Calculator Doolittle LU factorization with partial pivoting for a small real matrix. Same engine as Linear System. Not a CAS. Runs locally. › → Matrix Determinant Calculator Determinant of a small real matrix. Same engine as Linear System and the matrix workspace. Not a CAS. Runs locally. › → Matrix Eigenvalues Calculator Eigenvalues and eigenvectors of a small real matrix via the characteristic polynomial. Same engine as Linear System. Not a CAS. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear-algebra engine?

No. It is mode=leftnullspace on math.linear_algebra, the same engine as /calc/math/linear-system and /workspace/matrix. It reuses the RREF already used by /matrix-rref, on Aᵀ.

Is this the same as the right nullspace?

Only when A is symmetric. For [[1,2],[3,6]] the right kernel is [−2, 1] and the left kernel is [−3, 1]. Right kernel lives at /calc/math/nullspace.