Null space

The null space (also spelled nullspace) of an m by n matrix A is the set of all vectors v which are solutions to the equation:

Av = 0

It is also called the kernel of A if A is interpreted as a linear map.

In set notation, Null A = {v: v is in Rⁿ and Av = 0 }

The right singular vectors of A corresponding to zero singular values form an orthonormal basis for the null space of A. The dimension of this linear subspace is called the nullity of A. This can be calculated by the number of nonleading columns in the row echelon form of the matrix. The rank of any matrix plus its nullity equals the number of columns of that matrix - this is the Rank-Nullity Theorem.