The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0, a linear subspace of the space of all vectors. It is also called the kernel of A if A is interpreted as a linear map (see kernel). In set notation,

Null A = {v: v ∈ Rⁿ and Av = 0 }.

The dimension of this linear subspace is called the nullity of A. This can be calculated as the number of columns that don't contain pivots in the row echelon form of the matrix A. The Rank-Nullity Theorem states that the rank of any matrix plus its nullity equals the number of columns of that matrix.

The right singular vectors of A corresponding to zero singular values form an orthonormal basis for the null space of A.

The null space of A can be used to find and express all solutions (the complete solution) of the equation Av = b. If the free variables of v are set to zero, a solution to this reduced equation is called a particular solution. The complete solution of the equation is equal to the particular solution added to any linear combination of vectors from the null space. The particular solutions vary according to b, while the null space vectors do not.

To show this works we consider each direction. In one direction, if Av=b, and Au=0, as is the case for any linear combination u of vectors from the null space, then it's clear that A(v+u) = b. In the other direction, if we have a solution x to Ax=b, then we can find a vector in the null space with the same values for the free variables in x, and subtract it to find the corresponding particular solution. Such a vector must exist, since Au=0 has a solution for any fixed value of the free variables.

Here is a concrete example. Suppose x+y = 7. You can arbitrarily pick x as your pivot (particular) variable, leaving y as the free variable. To find the particular solution, we set all free variables (y) to zero, and solve to find that x = 7. To compute the nullspace, we set the free variable y to 1, change the constant to zero, and solve. This produces the equation x + 1 = 0, or x = -1. That means that the null space basis vector is [-1 1]. We conclude that the complete solution to x + y = 7 is [7 0] + c[-1 1], where c is an arbitrary real constant.