Find kernel vectors and understand constraints in seconds. Works with fractions or decimals, showing pivots. Download CSV and PDF reports for homework and research.
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The null space (kernel) of a matrix A is the set N(A) = { x ∈ ℝⁿ | A·x = 0 }.
This calculator computes the reduced row echelon form (RREF) of A using row operations. Pivot columns identify dependent variables; free columns become parameters.
The number of basis vectors equals the nullity: nullity(A) = n − rank(A).
The null space, or kernel, contains every vector x where A·x = 0. These vectors describe directions that leave the linear map unchanged. In data work, they reveal redundant features and hidden constraints. If A has n columns and rank r, then nullity n−r gives how many independent null directions exist. This calculator returns a basis so any solution is a linear combination of the displayed basis vectors.
To find that basis, the tool performs row reduction to reduced row echelon form (RREF). Leading ones identify pivot columns, which correspond to dependent variables. Remaining columns are free variables. For each free variable, the algorithm sets it to 1, sets other free variables to 0, and solves pivot entries from the RREF equations. In numeric mode, partial pivoting and a tolerance ε limit roundoff drift on nearly singular inputs.
Rank is the count of pivot columns and measures how many independent columns the matrix contains. Nullity is the dimension of the null space and equals the number of basis vectors produced. The identity nullity(A) = n − rank(A) provides an immediate consistency check beside the computed RREF. When nullity is zero, the only solution is x = 0, which means the columns are linearly independent over the underlying field.
Exact mode uses fraction arithmetic, simplifying each intermediate value, so results match textbook solutions and avoid rounding disputes. It accepts integers, decimals, or a/b entries, then reduces them to lowest terms. Numeric mode is faster for large values and noisy measurements, and the display precision controls rounding in the interface. Adjust ε when very small coefficients should be treated as zero after elimination, improving stability for repeated classroom verification.
After you submit, the calculator lists pivot and free column indices, prints the basis vectors, and can show the full RREF for auditing. The verification option multiplies A·v to confirm each basis vector returns a zero output within tolerance. Export CSV for spreadsheets, scripts, and grading, or export PDF for reports and assignments. A 10×10 limit keeps computation responsive on typical shared hosting with low memory overhead.
Use decimals in numeric mode. In exact mode, you can enter integers, decimals, or fractions like 7/12; the calculator simplifies them automatically.
The number of vectors equals nullity, which counts free variables in A·x = 0. Each free variable contributes one independent direction in the solution space.
Usually not. Any basis spanning the same null space is valid. Vectors can be scaled or combined while still representing the same solution set.
Choose exact mode for proofs, grading, or when you need simplified rational results. It avoids rounding and matches symbolic steps used in linear algebra courses.
In numeric mode, ε treats very small values as zero during elimination and cleanup. Increasing ε can reduce noise effects, while decreasing ε can preserve tiny but meaningful coefficients.
Then the matrix is full column rank, and the only solution to A·x = 0 is x = 0. The calculator will show no basis vectors and report nullity as 0.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.