This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

NullSpace

 NullSpace[m]gives a list of vectors that forms a basis for the null space of the matrix m.
• 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, ZeroTest->test] evaluates to determine whether matrix elements are zero.
• Possible settings for the Method option include , , and . The default setting of Automatic switches among these methods depending on the matrix given.
Find the null space of a 3×3 matrix:
The action of on the vector is the zero vector:
Find the null space of a 3×3 matrix:
 Out[2]=
The action of on the vector is the zero vector:
 Out[3]=
 Scope   (2)
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:
Find the null space symbolically:
 Options   (1)
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)
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:
is a 5×5 matrix:
The null space of :
Arbitrary linear combinations of the null space of give zero:
is a 3×4 matrix of random zeros and ones:
The MatrixRank equals the column dimension of minus the dimension of the null space: