# Null Space Calculator

Find the null space (kernel) of a 2x2 or 3x3 matrix. Shows basis vectors, nullity, rank, and determinant. Free online null space calculator.

## What this calculates

Find the null space of a 2x2 or 3x3 matrix. The null space is the set of all vectors x that satisfy Ax = 0. This calculator shows the basis vectors, nullity, rank, and determinant.

## Inputs

- **Matrix Size** — options: 2 x 2, 3 x 3 — Choose the dimension of your square matrix.
- **a₁₁ (Row 1, Col 1)**
- **a₁₂ (Row 1, Col 2)**
- **a₁₃ (Row 1, Col 3)** — Only used for 3x3 matrices.
- **a₂₁ (Row 2, Col 1)**
- **a₂₂ (Row 2, Col 2)**
- **a₂₃ (Row 2, Col 3)** — Only used for 3x3 matrices.
- **a₃₁ (Row 3, Col 1)** — Only used for 3x3 matrices.
- **a₃₂ (Row 3, Col 2)** — Only used for 3x3 matrices.
- **a₃₃ (Row 3, Col 3)** — Only used for 3x3 matrices.

## Outputs

- **Nullity (Dimension of Null Space)** — The number of free variables, i.e., the dimension of the null space.
- **Null Space Basis Vectors** — formatted as text — The basis vector(s) spanning the null space.
- **Rank** — The rank of the matrix (number of pivot columns).
- **Determinant** — The determinant of the matrix.

## Details

The null space (also called the kernel) of a matrix A is the set of all vectors x such that Ax = 0. It captures everything that the transformation "collapses" to zero.

**Key Definitions:**

- **Null space:** Null(A) = {x : Ax = 0}
- **Nullity:** The dimension of the null space (number of free variables)
- **Rank:** The number of pivot columns in row echelon form
- **Rank-Nullity Theorem:** rank + nullity = number of columns

**How to Find the Null Space:**

1. Write out the augmented matrix [A | 0]
2. Row reduce to reduced row echelon form (RREF)
3. Identify pivot columns and free columns
4. Express pivot variables in terms of free variables
5. Write the general solution as a linear combination of basis vectors

**Example:**

For A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], the determinant is 0 (rank 2), so the null space is one-dimensional. Row reducing gives one free variable and the null space basis vector (1, -2, 1).

**Why It Matters:**

The null space tells you about the solutions to Ax = b. If the null space is trivial ({0}), the system has at most one solution. If the null space is nontrivial, there are infinitely many solutions (you can add any null space vector to a particular solution).

## Frequently Asked Questions

**Q: What is the null space of a matrix?**

A: The null space is the set of all vectors x that get mapped to zero when multiplied by the matrix: Ax = 0. For an invertible matrix, the null space contains only the zero vector. For a singular matrix, the null space contains infinitely many vectors that form a subspace.

**Q: What is the relationship between rank and nullity?**

A: The Rank-Nullity Theorem states that rank + nullity = number of columns. For a 3x3 matrix, if the rank is 2, then the nullity is 1, meaning the null space is a one-dimensional subspace (a line through the origin). If the rank is 1, the null space is two-dimensional (a plane through the origin).

**Q: When is the null space trivial?**

A: The null space is trivial (contains only the zero vector) when the matrix has full rank, meaning its determinant is non-zero. In this case, Ax = 0 has only the obvious solution x = 0, and the matrix is invertible.

**Q: How is the null space used in practice?**

A: The null space is used in solving systems of equations (it determines the set of all solutions), image compression (SVD relies on null spaces), machine learning (identifying redundant features), and control theory (finding system states that produce zero output). It is fundamental in any application of linear algebra.

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Source: https://vastcalc.com/calculators/math/null-space
Category: Math
Last updated: 2026-04-08