Null Space Calculator - Find Matrix Kernel & Basis Vectors
Find the null space (kernel) of any matrix up to 4×4, calculate basis vectors, rank, and nullity using Gauss-Jordan elimination.
Select the matrix dimensions, fill in the entries, and click Calculate to find all basis vectors of the null space and the rank of the matrix.
Null Space Calculator - Find Matrix Kernel & Basis Vectors
Find the null space (kernel) of any matrix up to 4×4, calculate basis vectors, rank, and nullity using Gauss-Jordan elimination.
About the null space calculator
The null space of a matrix A (also called the kernel of A) is the set of all vectors x that satisfy the homogeneous equation Ax = 0. Geometrically, it is the set of all vectors that the linear transformation represented by A maps to the zero vector. The null space is always a subspace of the domain space, and its dimension is called the nullity of the matrix.
The rank-nullity theorem is one of the central results of linear algebra: for an m × n matrix A, rank(A) + nullity(A) = n. This means that rank and nullity together always add up to the number of columns. A matrix with full column rank (rank = n) has a trivial null space containing only the zero vector. When the rank is less than n, the null space has positive dimension equal to n − rank, and there are infinitely many vectors satisfying Ax = 0.
To compute the null space, this calculator uses Gauss-Jordan elimination to reduce A to its Reduced Row Echelon Form (RREF). In RREF, each non-zero row has a leading 1 (pivot) and every other entry in that column is zero. The columns containing pivots correspond to basic variables; the remaining columns correspond to free variables. For each free variable, one can set it to 1 and all other free variables to 0, then back-substitute to find the values of the basic variables. The resulting vector is a basis vector of the null space.
The null space has important applications throughout applied mathematics and engineering. In linear systems of equations, the null space tells you about the non-uniqueness of solutions: if Ax = b has one solution x₀, then the general solution is x₀ plus any element of the null space. In control theory, the null space of a controllability matrix reveals uncontrollable modes. In signal processing, the null space of a measurement matrix identifies signals that are invisible to the sensor array. In chemistry, the null space of the stoichiometric matrix gives all conservation laws of a reaction network.
For numerical stability, this calculator uses partial pivoting during Gaussian elimination and treats any value smaller than 1e-10 in absolute value as zero. This makes the algorithm robust for matrices with integer or rational entries encountered in typical coursework and engineering problems. Enter any numbers — integers, decimals, or fractions expressed as decimals — and the calculator immediately returns the rank, nullity, and a complete set of null space basis vectors.
Null space examples
Four examples spanning different matrix shapes and null space dimensions.
| Matrix | Null Space Basis | Explanation |
|---|---|---|
| 2×3: [[1,2,3],[4,5,6]] | v1 = [1, −2, 1] | Rank 2, nullity 1. One free variable produces one basis vector. Verify: 1·1 + 2·(−2) + 3·1 = 0 and 4·1 + 5·(−2) + 6·1 = 0. |
| 3×3 Identity [[1,0,0],[0,1,0],[0,0,1]] | Trivial (zero vector only) | Full rank matrix: rank = 3, nullity = 0. The only solution to Ix = 0 is x = 0. |
| 3×3 rank-deficient: [[1,2,3],[2,4,6],[1,1,2]] | v1 = [−1, −1, 1] | Rank 2, nullity 1. Rows 1 and 2 are linearly dependent (row 2 = 2×row 1). RREF yields pivot columns 0 and 1, free column 2; back-substitution gives v = [−1, −1, 1]. |
| 2×2 zero matrix [[0,0],[0,0]] | v1 = [1,0], v2 = [0,1] | Rank 0, nullity 2. Every vector satisfies Ax = 0, so the entire R² is the null space with standard basis. |
How to use the null space calculator
- Select the matrix dimensions (rows × columns) using the size buttons. Available sizes range from 2×2 to 4×4, including non-square matrices like 2×3 and 3×4.
- Enter the matrix entries in the grid. Each cell accepts any real number including decimals and negatives. Leave blank cells will trigger an error.
- Click Calculate Null Space. The result shows the rank, nullity, and all basis vectors of the null space.
- Use the Load Example buttons to pre-fill classic examples: a 2×3 matrix with one-dimensional null space, or a rank-deficient 3×3 matrix.
- Click Reset to clear all cells while keeping the current matrix size, or change the size selector to start fresh with a different matrix dimension.
Null space calculator FAQ
What is the null space of a matrix?
The null space of a matrix A is the set of all vectors x such that Ax equals the zero vector. It represents all directions in the input space that the linear transformation A collapses to zero. The null space is always a subspace (it contains the zero vector, and is closed under addition and scalar multiplication). Its dimension, called the nullity, measures how much information A loses during the transformation.
How does Gauss-Jordan elimination find the null space?
The algorithm converts A to Reduced Row Echelon Form (RREF) by applying row operations. In RREF, pivot columns and free columns are easily identified. For each free variable (non-pivot column), setting that variable to 1 and all others to 0, then solving back for the pivot variables, yields one null space basis vector. The complete null space is the span of all such vectors.
What does it mean if the null space is trivial?
A trivial null space contains only the zero vector. This happens when the matrix has full column rank — every column is a pivot column and there are no free variables. For the equation Ax = 0, the unique solution is x = 0. A square matrix with trivial null space is invertible; a non-square matrix with trivial null space has the equation Ax = b with at most one solution for any b.
What is the rank-nullity theorem?
The rank-nullity theorem states that for an m × n matrix A: rank(A) + nullity(A) = n, where n is the number of columns. Rank is the dimension of the column space (number of linearly independent columns), and nullity is the dimension of the null space. They are complementary: increasing rank decreases nullity and vice versa. This theorem is fundamental to understanding linear maps and systems of equations.
Can a non-square matrix have a null space?
Yes. Any matrix where the number of columns exceeds the rank has a non-trivial null space. For a wide matrix with more columns than rows (m < n), the rank can be at most m, so nullity ≥ n − m > 0, guaranteeing a non-trivial null space. Tall matrices (more rows than columns) can still have a trivial null space if their columns are linearly independent.
Why might the basis vectors contain decimal values?
When the matrix has non-integer entries or when back-substitution produces fractions, the null space basis vectors will have decimal components. This is mathematically correct — the null space is defined over the real numbers, not just integers. You can scale any basis vector by any non-zero constant and get an equally valid basis vector, so if you prefer integer components, multiply the vector by the LCM of its denominators.