Last updated October 5, 2026 · Report an error
The null space (or kernel) of a matrix A is the set of all vectors x that satisfy Ax = 0 — it reveals whether a matrix is singular and, if so, how many independent directions it collapses to zero.
This calculator finds the null space basis vectors and nullity of any matrix — square or rectangular — with rows and columns each from 2 up to 6.
How it works
Enter the number of rows and columns and the matrix values — they don't need to match. The calculator reduces the matrix to reduced row echelon form, identifies which columns lack a pivot (the free variables), and constructs one basis vector for each free variable by back-solving the pivot rows.
- Click Calculate to see your results.
Examples
Singular 2×2 example
The matrix [[1,2],[2,4]] (whose rows are proportional) has nullity 1, with a null space basis vector of (-2, 1).
Identity matrix
The 2×2 identity matrix has nullity 0 — its only solution to Ax = 0 is the trivial zero vector, since the matrix is full rank.
Who should use it
- Checking linear algebra coursework on null space, kernel, and singular matrices.
- Determining whether a matrix is invertible before attempting to compute its inverse.
Industry applications
- Linear algebra education
- Engineering, physics, and applied mathematics
Advantages
- Automatically identifies free variables and constructs a full basis, not just the nullity number.
- Handles both square and rectangular matrices, up to 6 rows and 6 columns.
Limitations
- Capped at 6 rows and 6 columns to keep computation instant on a public, synchronous endpoint.
Common mistakes to avoid
- Expecting a non-trivial null space for every matrix — full-rank (non-singular) matrices only have the zero vector.
- Forgetting that each returned vector is only one basis vector — any scalar multiple (or combination, if nullity > 1) is also in the null space.
Best practices
- Verify a returned basis vector by multiplying it against the original matrix — the result should be the zero vector.
Tips
- A matrix has an inverse if and only if its null space is trivial (nullity 0) — this calculator doubles as a quick singularity check.