Enter your matrix dimensions and values to find the null space (kernel) and nullity of any matrix. Set the number of rows and columns (up to 4×4), fill in the matrix entries, and get back the basis vectors for the null space and the nullity dimension — computed via row reduction.
Nullity (Dimension of Null Space)
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Rank of Matrix
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Matrix Size
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Null Space Basis Vectors
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The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It represents all solutions to the homogeneous equation and is a subspace of the domain of A. In practice, finding it means identifying every vector that gets mapped to the zero vector by the linear transformation defined by A.
Nullity is the dimension of the null space — it counts how many linearly independent vectors form the basis of the null space. The Rank-Nullity Theorem states that rank(A) + nullity(A) = number of columns of A. So if your matrix has 4 columns, rank 3 means nullity is 1.
First, set up the equation Ax = 0. Then reduce the matrix A to its reduced row echelon form (RREF). Identify the free variables (columns without pivot positions). Express the pivot variables in terms of the free variables, then write the general solution as a linear combination of basis vectors — those vectors form the null space basis.
The null space contains only the zero vector when the matrix has full column rank — meaning every column has a pivot and there are no free variables. This happens when rank(A) equals the number of columns, making nullity = 0. For square matrices, this is equivalent to the matrix being invertible.
The null space of A contains all x such that Ax = 0 (right null space). The left null space of A contains all y such that A^T y = 0, which is the null space of the transpose. They are different subspaces associated with the same matrix and generally have different dimensions.
Yes. Any matrix with more columns than rows (a 'wide' matrix) must have a non-trivial null space, since it cannot have full column rank. Even square or tall matrices can have non-trivial null spaces if their columns are linearly dependent. The nullity equals columns minus rank.
The null space reveals the structure of a linear system. It tells you whether Ax = b has a unique solution (trivial null space), infinitely many solutions (non-trivial null space), and it underpins concepts like linear independence, basis, and the fundamental theorem of linear algebra.
Enter your matrix dimensions (up to 4×4) and fill in the entries. The calculator performs Gaussian elimination to reduce the matrix to row echelon form, identifies pivot and free columns, then back-substitutes to express the basis vectors of the null space. It also reports the rank and nullity.