The null space of a matrix A is every vector x with A·x = 0. Reduce A to reduced row echelon form, set each free variable to 1 in turn and read off the pivot variables. For [[1,2,3],[4,5,6],[7,8,9]] the rank is 2, the nullity is 1, and the null space is spanned by (1, -2, 1).
Students checking homework, engineers looking for dependent columns and anyone solving a homogeneous system need the same thing: a basis for the set of x that A sends to zero. Type the matrix one row per line and the calculator row-reduces it with exact fractions, so no rounding noise creeps into the pivots. It reports the rank, the nullity (rank plus nullity always equals the number of columns), each basis vector scaled to whole numbers, and every row operation it performed. The basis it returns is the standard one built from free variables; any other basis of the same space is equally correct, so a textbook answer may look different while spanning the same subspace. Matrices are limited to 8 rows and 8 columns.
Put each row on its own line and separate entries with spaces or commas. Fractions such as -3/4 and decimals such as 0.5 are accepted.
The headline is the nullity, the number of free variables. Rank is shown beside it; the two add up to the column count.
The table lists one vector per free variable, multiplied through so its entries are coprime integers.
The working lists each row exchange, scaling and elimination, then the final RREF. Compare it line by line with your own reduction.
Result Nullity (dimension of the null space): 1
Rank: 2
Matrix size: 3 × 3
Null space basis: (1, -2, 1)
Row three is twice row two minus row one, so the rank is 2 and one free variable gives the single basis vector (1, -2, 1).
Result Nullity (dimension of the null space): 2
Rank: 1
Matrix size: 1 × 3
Null space basis: (-1, 1, 0); (-1, 0, 1)
A single equation in three unknowns leaves two free variables, so the null space is a plane spanned by two vectors.
Result Nullity (dimension of the null space): 0
Rank: 2
Matrix size: 2 × 2
Null space basis: Only the zero vector
Every column holds a pivot, so the only solution of A·x = 0 is the zero vector and the nullity is 0.
N(A) = { x : A·x = 0 }, nullity = n − rank(A)
Axrank(A)nGauss-Jordan elimination turns A into reduced row echelon form without changing its null space. Columns with a leading 1 are pivot variables; the rest are free. For each free variable, set it to 1 and the other free variables to 0, then each pivot variable equals minus the RREF entry in that free column. The resulting vectors are linearly independent and span the null space. All arithmetic uses exact fractions, and each vector is then multiplied by the least common denominator and divided by the common factor.
Nothing in practice. Kernel is the name used for the linear map x ↦ A·x, null space the name used for the matrix. Both mean the set of vectors that map to the zero vector.
A subspace has infinitely many bases. A book might pick (-1, 2, -1) where this calculator shows (1, -2, 1), or combine two basis vectors differently. Check that each of your vectors satisfies A·x = 0 and that you have as many independent vectors as the nullity.
Every column has a pivot, so the columns are linearly independent and A·x = 0 has only the trivial solution x = 0. For a square matrix this means A is invertible and its determinant is non-zero.
For an m × n matrix, rank plus nullity equals n. The calculator counts pivot columns for the rank and free columns for the nullity, so the two numbers it shows always add up to the number of columns you entered.
Yes. Entries like 3/4, -2/5, 0.25 or -1.5 are converted to exact fractions before elimination, so the reduction has no rounding error. Decimals are limited to six places.
After back-substitution a vector may contain fractions such as (-3/2, 1). Multiplying by the least common denominator gives (-3, 2), which spans the same line and is easier to read and check. The working shows the unscaled vector too.
No. Swapping rows, scaling a row by a non-zero number and adding a multiple of one row to another all preserve the solution set of A·x = 0, which is why reducing to RREF is a valid way to find it.
Using this tool in an assignment, article or report? Copy a citation, or a link for your website.
Tools Directory Editorial Team. (2026, October 24). Null Space Calculator. Tools Directory. Retrieved October 26, 2026, from https://tools.milunir.com/math/algebra-tools/null-space-calculator/
Tools Directory Editorial Team. “Null Space Calculator.” Tools Directory, 24 Oct. 2026, https://tools.milunir.com/math/algebra-tools/null-space-calculator/. Accessed 26 Oct. 2026.
Tools Directory Editorial Team. “Null Space Calculator.” Tools Directory. Last modified October 24, 2026. Accessed October 26, 2026. https://tools.milunir.com/math/algebra-tools/null-space-calculator/.
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