# NullSpace

NullSpace[m]

gives a list of vectors that forms a basis for the null space of the matrix m.

# Details and Options • NullSpace works on both numerical and symbolic matrices.
• The following options can be given:
•  Method Automatic method to use Modulus 0 integer modulus to use Tolerance Automatic numerical tolerance to use ZeroTest Automatic function to test whether matrix elements should be considered to be zero
• NullSpace[m,Modulus->n] finds null spaces for integer matrices modulo n.
• NullSpace[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero.
• Possible settings for the Method option include "CofactorExpansion", "DivisionFreeRowReduction", and "OneStepRowReduction". The default setting of Automatic switches among these methods depending on the matrix given.

# Examples

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## Basic Examples(1)

Find the null space of a 3×3 matrix:

The action of m on the vector is the zero vector:

## Scope(10)

### Basic Uses(6)

Null space of a machine-precision matrix:

Null space of a complex matrix:

Null space of an exact matrix:

Find the null space symbolically:

Null space of non-square matrices:

The null space of a large numerical matrix is computed efficiently:

### Special Matrices(4)

Null space of a sparse matrix:

Null space of structured matrices:

always has an empty null space::

The null space of IdentityMatrix[{m,n}] is non-empty:

Compute the null space for HilbertMatrix:

## Options(1)

### Modulus(1)

m is a 3×3 random matrix of integers between 0 and 4:

Use arithmetic modulo 5 to compute the null space:

The vector is in the null space modulo 5:

## Applications(2)

m is a 3×3 singular matrix with a nonempty null space:

Find a solution for :

All solutions are given by , where is any vector in the null space:

Find a basis for the eigenspace for a particular eigenvalue:

## Properties & Relations(2)

m is a 5×5 matrix:

The null space of m:

Arbitrary linear combinations of the null space of m give zero:

m is a 3×4 matrix of random zeros and ones:

The MatrixRank equals the column dimension of m minus the dimension of the null space: