NullSpace

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

Details and Options

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.

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Find the null space of a 3×3 matrix:

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

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:

Null space of a sparse matrix:

Null space of structured matrices:

IdentityMatrix[n] always has an empty null space::

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

Compute the null space for HilbertMatrix:

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:

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:

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:

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