NullSpace

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 (2)

m is a 3×4 matrix:

Use exact arithmetic to find the null space:

Use machine arithmetic:

Use 20-digit precision arithmetic:

Compute the null space for a complex matrix:

Generalizations & Extensions (1)

Find the null space symbolically:

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:

Introduced in 1988

(1.0)

Updated in 1996

(3.0)

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