Null Space Calculator
The null space calculator parses integer or fractional matrices, computes exact RREF without floating-point tolerance, constructs basis vectors and verifies every vector against the original matrix.
Null Space Calculator inputs
The null space calculator parses integer or fractional matrices, computes exact RREF without floating-point tolerance, constructs basis vectors and verifies every vector against the original matrix.
Separate entries by spaces or commas and rows by new lines. Fractions such as 2/3 are supported. Maximum 10 rows by 12 columns.
Null Space Calculator results
How to use the null space calculator
The null space calculator parses integer or fractional matrices, computes exact RREF without floating-point tolerance, constructs basis vectors and verifies every vector against the original matrix. The sections below explain the data, assumptions and limitations so the answer can be checked rather than accepted as a generic black-box output.
Matrix input and exact parsing
The null space calculator accepts rows on separate lines and entries separated by spaces or commas. Integer and fraction forms are converted to exact numerator-denominator values. Each row must have the same number of columns. This avoids the hidden tolerance choices that can cause a nearly singular decimal matrix to receive different ranks in different software.
Reduced row echelon form
Gaussian elimination locates a nonzero pivot, swaps it into position, scales the pivot to one and eliminates that column from every other row. The process advances across the columns to produce reduced row echelon form. Pivot columns and free columns are recorded from the actual elimination, not guessed from zero columns in the original matrix.
Constructing the basis
For each free variable, the null space calculator sets that variable to one and all other free variables to zero. Pivot variables are then obtained from the RREF equations with opposite signs. Each resulting vector is one basis direction. Any solution of Ax = 0 is a linear combination of those basis vectors.
Rank-nullity theorem
Rank is the number of pivots, nullity is the number of free columns, and rank plus nullity equals the number of matrix columns. The result cards show this identity explicitly. A square full-rank matrix has nullity zero and only the zero vector in its null space; a matrix with dependent columns has one or more nonzero basis vectors.
Verification against the original matrix
After construction, every basis vector is multiplied by the original matrix, not only the RREF. The products must be exact zero fractions. This check can detect a sign or indexing error in the basis construction. The page reports the verification in prose and keeps the exact fractions for inspection.
Scope and numerical matrices
The browser solver is limited to a practical matrix size and exact rational entries. Large sparse systems, symbolic parameters and measured decimal data may be better handled by specialist linear-algebra software with a documented tolerance. The null space calculator is especially useful for coursework, small models and checking a hand-reduced matrix without rounding ambiguity.
Null Space Calculator questions
What does nullity mean?
It is the dimension of the null space and equals the number of free variables.
Why are fractions kept exact?
Exact arithmetic avoids rank changes caused only by floating-point rounding tolerance.
What if nullity is zero?
The only solution to Ax = 0 is the zero vector.