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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 Engine tested · Specification checked · 44/44 tests · Production surface contract 3/3 · v1.10.0
Input interpretation
Enter values to calculate.
Result
—
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 · 44/44 tests · Production surface contract 3/3 · Specification checked · v1.10.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
23 golden · 2 boundary · 19 property · Production surface contract 3/3 · Artifact integrity PASS
Production
Embedded snapshot: STALE · Last attested schema matched 1.10.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · Public/cache ✓ · Origin ✓ · Live production status STALE (1 capability; 163 remain CURRENT) @ 2026-09-19T00:00:17.039Z
Semantic contract
PASS

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 cutoff 10⁻¹⁵.

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 1.10.0 · 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
  • 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 · 44/44 tests · Production surface contract 3/3 · 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
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.10.0

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

→ 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 square system. Full rank uses QR; rank-deficient uses SVD. Same engine as Linear System. Not Statistics regression. Runs locally. › → Linear Equation Calculator Solve ax + b = rhs for x. Identity and inconsistent cases when a = 0. Not the quadratic formula. Runs locally. › → Linear System Calculator Solve small linear systems Ax = b, plus det, inverse, rank, LU, 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. ›
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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.