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 · Specification checked · v1.14.12 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.14.12 · CVP protocol 1.0.0-proposed · Evidence 2026-10-09.qr-root-position-disks
Verification revision
2026-10-09.qr-root-position-disks · 10/10 property · digest 0214764e3fff
Legacy regression
58/58 tests · Production surface contract 5/5
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 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 10/10 (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.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
431/431 golden · 8/8 CVP boundary · 68/68 invalid · 10/10 property · 1/1 metamorphic · 48/48 cross-interface · 2/2 cross-calculator · 5/5 CVP contract · Manifest
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
29 legacy golden · 6 legacy boundary · legacy regression suite · 431/431 oracle-backed golden · 68/68 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.14.12 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 1.14.0 · 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
  • 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 5/5 · 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
1.14.0

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.