Enter a matrix to compute its kernel
The solver finds all vectors x such that Ax = 0. It also reports rank, nullity, pivots, free variables, and a basis.
Example data table
| Example matrix | Rank | Nullity | Kernel basis |
|---|---|---|---|
|
[1 2 3 4] [2 4 6 8] [0 1 1 1] |
2 | 2 | (-1, -1, 1, 0), (-2, -1, 0, 1) |
|
[1 0 0] [0 1 0] [0 0 1] |
3 | 0 | Trivial kernel only |
Formula used
Kernel definition: Ker(A) = {x ∈ Rn : Ax = 0}
Rank-nullity relation: nullity(A) = n - rank(A)
Basis construction: Solve the homogeneous system after reducing A to reduced row echelon form. Free variables create kernel basis vectors.
The calculator row-reduces the matrix to identify pivot columns and free columns. Each free variable is assigned 1 once and 0 otherwise, then pivot variables are solved from the reduced system.
This produces a basis for the null space. If no free variables exist, the kernel is trivial and contains only the zero vector.
How to use this calculator
- Select the number of rows and columns for your matrix.
- Enter matrix values in the generated input grid.
- Adjust display decimals and pivot tolerance if needed.
- Click Calculate Kernel to solve Ax = 0.
- Review rank, nullity, pivot columns, free variables, and basis vectors.
- Use the CSV or PDF buttons to save the result summary.
FAQs
1. What does the kernel of a matrix mean?
The kernel is the set of all vectors that become the zero vector after multiplication by the matrix. It is also called the null space.
2. What is a trivial kernel?
A trivial kernel means the only solution to Ax = 0 is the zero vector. This happens when every column variable is a pivot variable.
3. Why does nullity matter?
Nullity tells you the dimension of the kernel. It shows how many independent directions map to zero under the transformation.
4. How are basis vectors found?
After row reduction, the solver identifies free variables. Each free variable generates one basis vector by setting it to 1 and the others to 0.
5. Can I enter decimal values?
Yes. The matrix accepts integers and decimals. The tolerance option helps classify tiny values as zeros during row reduction.
6. Why are rank and nullity shown together?
They are linked by the rank-nullity theorem. For an n-column matrix, nullity equals n minus the rank.
7. What if the matrix is rectangular?
That is valid. The calculator handles rectangular matrices and still solves the homogeneous system using reduced row echelon form.
8. Does exporting save my full matrix result?
Yes. The export summary includes matrix size, kernel type, rank, nullity, pivot columns, free variables, and basis vectors.