Null Space Calculator - Basis, Rank, and Nullity

A null space calculator finds every vector x that a given matrix A sends to the zero vector, that is, every solution of A*x = 0. The set of all such vectors is the null space (or kernel) of A, written N(A) or ker(A). Type the nine entries of a 3x3 matrix and the page returns the nullity, the rank, and a basis of N(A).

• • Linear algebra homework: Solve a problem that asks for the basis of N(A), the nullity, or the rank of a 3x3 matrix.
• • Check whether A is invertible: When the nullity is 0, the only solution to A*x = 0 is x = 0, so det(A) is non-zero and the inverse exists.
• • Differential equations and physics: Find the null space of a coefficient matrix to build the homogeneous solution of a linear ODE or to identify degenerate directions in a physical model.
• • Graphics and robotics: Compute the directions in which a transformation collapses space, the geometric meaning of N(A) for a singular transform matrix.

The null space is one of the four fundamental subspaces of a matrix. The page focuses on N(A) because it pairs with A*x = 0, the system-of-equations page solves for non-zero right-hand sides.

The returned basis is also a sanity check. Any vector in N(A) is a linear combination of the listed basis vectors (with scalars, since the null space is a subspace), so if a vector you wrote down by hand cannot be written that way, it is not in N(A).

The null space is one of the four fundamental subspaces associated with any matrix, and the System of Equations Calculator page solves the related non-homogeneous system A*x = b that lives in the same row-reduced framework.

The page reads the nine numeric inputs as a 3x3 matrix A, row-reduces A to reduced row echelon form (RREF), and reads the null space off the free columns of that RREF.

• A: 3x3 input matrix with entries a_11 through a_33.
• x: Unknown column vector (x1, x2, x3) that solves A*x = 0.
• rank(A): Number of pivot rows in RREF(A); also the number of linearly independent rows (equivalently columns) of A.
• nullity(A): Dimension of N(A); equals 3 - rank(A) for a 3x3 matrix by the rank-nullity theorem.
• Free variables: Columns without a leading 1 in RREF(A); each one produces one basis vector of N(A).

Once the RREF is in hand, the basis vectors of N(A) fall out one per free column. For each free column, set that free variable to 1 with the other free variables at 0, then read the pivot variables off the RREF as their negatives; the full set of those directions is the basis.

According to Wolfram MathWorld, the null space of a matrix A is the kernel of the linear map x -> A*x, and its dimension (the nullity) satisfies rank(A) + nullity(A) = n, where n is the number of columns of A.

If the calculator reports nullity = 0, the matrix is full rank and the Multiplicative Inverse Calculator page computes A^{-1} from the same RREF, so the two answers are the same computation finished differently.

Four ideas explain why the null space is what it is and how the calculator's output should be read.

These four ideas are the building blocks of the rest of linear algebra. The null space of A is dual to the column space, so the same RREF gives you most of the other fundamental subspaces. The characteristic polynomial page studies the null space of A - lambda*I, the eigenspace for eigenvalue lambda.

If the returned basis looks unfamiliar, the vector-magnitude page gives the length of each basis vector, and the dot-product page returns the dot product of any two basis vectors of N(A), with a result of 0 confirming orthogonality between them.

If you want the length of each basis vector the null space calculator returns, the Vector Magnitude Calculator page takes the dot product of a vector with itself and takes the square root, all from a single 3-component input.

Five short steps walk you from the original matrix to a verified basis.

1. 1 Enter row 1 of A: Type a_11, a_12, and a_13. The default (1, 2, 3) starts on the classic rank-1 null space problem.
2. 2 Enter row 2 of A: Type a_21, a_22, and a_23. The default (2, 4, 6) is twice row 1 and forces rank = 1.
3. 3 Enter row 3 of A: Type a_31, a_32, and a_33. The default repeats row 1, giving the same null space as the rank-1 case.
4. 4 Read the rank and nullity: Nullity is the primary output, with rank below. They always add up to 3 for a 3x3 matrix.
5. 5 Read the basis of N(A): The basis row lists one vector per free variable. Nullity 0 means only (0, 0, 0); nullity 3 means the standard basis e1, e2, e3.
6. 6 Reset or change size: Click Reset to restore the example. For 2x2 systems or non-square A, use the system-of-equations page in a new tab.

Try A = [[1, 0, -1], [0, 1, 2], [1, 1, 1]]. Pivots are columns 1 and 2, free column is 3, and setting x3 = 1 gives x1 = 1, x2 = -2. The calculator reports rank = 2, nullity = 1, basis: v1 = (1, -2, 1).

The natural follow-up for a rank-deficient matrix is to study the null space of A - lambda*I instead, and the Characteristic Polynomial Calculator page returns the eigenvalues where that null space becomes non-trivial.

These benefits matter most when the hand calculation is error-prone but the result has to be read off the RREF by hand.

• • Skip the row-reduction bookkeeping: The page handles the row swaps, pivot scaling, and column clearing, the steps most likely to introduce mistakes in a hand calculation.
• • Get rank, nullity, and basis in one pass: All three answers come from the same RREF, so the page returns them together and lets you read the rank-nullity check from a single line.
• • See the null space as a list of basis vectors: The page returns it as a list of triples that you can paste into an answer or script.
• • Sanity check for a hand-computed basis: If a vector you wrote down by hand is not a linear combination of the returned basis vectors, the page makes the mismatch visible right away.
• • Connect to the rest of linear algebra: If your next step is the eigenspace, the inverse, or the column space, the page links to the peer calculator that does each one.

The page is most useful as a check, not as a replacement for understanding the RREF. Use it to confirm a homework answer or pre-compute a null space before handing the matrix to a longer script.

For a 3x3 matrix, N(A) can be a point, a line, a plane, or the whole space. The rank-nullity line at the bottom of the result panel is a quick consistency check that rank + nullity = 3.

If you next need the classical adjoint of A to compute A^{-1} by hand, the Adjoint Matrix Calculator page builds the cofactor matrix that the multiplicative-inverse formula calls for, and the same null space output is what tells you whether the inverse exists at all.

The algorithm is the same in every case, but a few factors change what the result looks like.

• • The page is fixed to 3x3 matrices. For 2x2, 4x4, or non-square m x n systems with m != n, use the system-of-equations page or a general linear-algebra tool.
• • Rounding the matrix entries to a small number of decimal places can flip a near-singular matrix from full rank to rank deficient, which makes the null space larger than the exact answer would suggest.
• • The page returns a basis of the null space, not the row space or the column space. The column space has dimension equal to the rank, and the row space is the column space of A transposed.

According to Encyclopaedia Britannica, the null space of a matrix A is the set of all vectors x such that A times x equals the zero vector, and it is also called the kernel of A.

If you want to check whether two basis vectors of N(A) are orthogonal, the Dot Product Calculator page returns the dot product of any two 3-component vectors, and a result of 0 confirms it.