Nullraums

Nullraums, called null spaces in English, are subspaces of a vector space consisting of all vectors that map to the zero vector under a given linear transformation. For a matrix A, the nullspace Null(A) = { x in F^n | A x = 0 }. It is the kernel of the associated linear map T(x) = A x. The term Nullraum is German; the plural is Nullräume.

This set is a subspace of the domain, containing the zero vector and closed under addition and scalar multiplication. Its dimension, called the nullity of A, measures the number of independent directions of solutions. The Rank-Nullity Theorem states that dim(domain) = rank(A) + nullity(A).

Computation is typically by solving the homogeneous system A x = 0. Row-reducing A to reduced or row echelon form yields parametric equations in terms of free variables; each free variable yields a basis vector for the nullspace. Example: For A = [[1,0,2], [0,1,-3]], Null(A) = span{(-2,3,1)}.

Applications include solving homogeneous systems, determining linear dependencies among columns, and studying linear operators in differential equations and control theory. If A is square and invertible, the nullspace is {0}. Related concepts include the left null space and the column space, via the rank-nullity relation.