Null space of a machine-precision matrix:

Null space of a complex matrix:

Null space of an exact matrix:

Null space of an arbitrary-precision 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 nonempty:

Compute the null space for HilbertMatrix:

Compute the null space of a matrix of univariate polynomials of degree :

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:

The following three vectors are not linearly independent:

Therefore the null space of the matrix whose rows are the vectors is nonempty:

The following three vectors are linearly independent:

Therefore the null space of the matrix whose rows are the vectors is empty:

Determine if the following vectors are linearly independent or not:

The matrix formed from the vectors has a nonempty null space, so they are not linearly independent:

Find the dimension of the column space of the following matrix:

Since the null space is empty, the dimension of the column space equals the number of columns:

Find the dimension of the subspace spanned by the following vectors:

Since the matrix rank of the matrix formed by the vectors is three, that is the dimension of the subspace:

Determine if the following system of equations has a unique solution:

Rewrite the system in matrix form:

The coefficient matrix has an empty null space, so the system has a unique solution:

Verify the result using Solve:

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:

Determine if the following matrix has an inverse:

The null space is not trivial, so the matrix is not invertible:

Verify the result using Inverse:

Determine if the following matrix has a nonzero determinant:

Since the null space is empty, its determinant must be nonzero:

Confirm the result using Det:

is an eigenvalue of if the null space of is nontrivial. A matrix is deficient if it has an eigenvalue whose multiplicity is greater than the dimension of the null space of . Show that is an eigenvalue for the following matrix :

Confirm the result using Eigenvalues:

The matrix is deficient because 2 appears twice, but eigenspace is one-dimensional:

Confirm the result with Eigensystem, which indicates deficiency by padding the eigenvector list with zeros:

Find a basis for the eigenvectors of the following matrix:

First, compute the eigenvalues:

For each unique eigenvalue, find the null space:

Combine the two one-dimensional spaces and one two-dimensional space using Join and Apply:

Confirm the result using Eigenvectors:

Estimate the fraction of random 10×10 0–1 matrix that are invertible: