Null Space Calculator: Kernel Basis, Rank & Nullity

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Formula at a glance

Null Space Calculator: Null space — N(A) = { x : Ax = 0 }. Example: Rank-2 3 × 3 matrix → Null space basis: (1, −2, 1) — How the Null Space Calculator works: Null space — N(A) = { x : Ax = 0 }

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📘 About the Null Space

The null space (kernel) of a matrix A is the set of all vectors x with Ax = 0. Its dimension, the nullity, plus the rank equals the number of columns (rank–nullity theorem). A non-trivial null space means the columns are dependent and Ax = b has infinitely many solutions when it has any.

Enter a matrix. The calculator row-reduces it with exact fractions, identifies pivot and free variables, writes each pivot variable in terms of the free ones, and gives one basis vector per free variable, scaled to whole numbers.

🧮 Formula

Null space

N(A) = { x : Ax = 0 }

Rank–nullity

rank(A) + nullity(A) = number of columns

✏️ Worked examples

Rank-2 3 × 3 matrix

You enter
Matrix A	1 2 3; 2 4 6; 1 1 1

You get
Null space basis	(1, −2, 1)
Nullity (dimension of the null space)	1
Rank	2
Rank–nullity check	2 + 1 = 3 columns ✓
RREF(A)	[1, 0, −1; 0, 1, 2; 0, 0, 0]

Basis (1, −2, 1).

2 × 4 matrix

You enter
Matrix A	1 0 2 -1; 0 1 -1 3

You get
Null space basis	(−2, 1, 1, 0), (1, −3, 0, 1)
Nullity (dimension of the null space)	2
Rank	2
Rank–nullity check	2 + 2 = 4 columns ✓
RREF(A)	[1, 0, 2, −1; 0, 1, −1, 3]

Basis (−2, 1, 1, 0), (1, −3, 0, 1).

🧭 How to use this calculator

What you enter

Matrix ARows separated by ; or new lines, entries by spaces or commas. Fractions like 1/2 are fine.

What you get

Null space basis
Nullity (dimension of the null space)
Rank
Rank–nullity check
RREF(A)

❓ Frequently asked questions

How do you find the null space?

Row-reduce A to RREF, write the pivot variables in terms of the free variables, and set each free variable to 1 in turn.

What if the null space is only {0}?

Then the columns are linearly independent and Ax = b has at most one solution.

Is the null space the same as the kernel?

Yes — the kernel of the linear map x ↦ Ax.

N(A) Null Space (Kernel) Calculator