---
title: "Null Space & Column Space Calculator — Basis with RREF Steps"
description: "Find the null space (kernel) and column space basis of any matrix using RREF. Shows rank, nullity, pivot columns, free columns, and basis vectors. Essential for linear algebra and differential equations."
url: "https://worldcalculators.org/calculators/null-space/"
canonical: "https://worldcalculators.org/calculators/null-space/"
---

# Null Space & Column Space Calculator

Find the null space (kernel) and column space basis of any matrix using RREF. Shows rank, nullity, pivot columns, free columns, and basis vectors. Essential for linear algebra and differential equations.

## Quick answer

Matrix A ( 3 × 4 )

## The Four Fundamental Subspaces

Gilbert Strang's "four fundamental subspaces" framework shows how an m×n matrix A creates four orthogonally complementary subspaces that together describe everything about the linear map A represents:

## Frequently Asked Questions

Ax = b has a solution iff b lies in the column space of A. If b is not in C(A), the system is inconsistent (no solution). If b is in C(A) and the null space is trivial 0, the solution is unique. If the null space has positive dimension (nullity > 0), there are infinitely many solutions: x = x_particular + any vector in N(A).
"Null space" and "kernel" are the same thing — different terminology from different traditions. US linear algebra texts (Strang, Anton) typically say "null space." European functional analysis texts say "kernel" (German: Kern, French: noyau). Both mean the set of all x satisfying Ax = 0. This calculator uses both terms interchangeably.

Source: https://worldcalculators.org/calculators/null-space/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.