Kernbasis

Kernbasis, or kernel basis, is a term used in linear algebra to describe a basis for the kernel (also called the null space) of a linear map or matrix. The kernel consists of all vectors that map to zero under the given linear transformation.

For a matrix A in a field F with size m by n, the kernel is the set of all vectors x in F^n such that Ax = 0. The kernel is a subspace of F^n, and its dimension is called the nullity of A. The kernel is sometimes referred to as the solution space to the homogeneous system Ax = 0.

Computing a kernbasis typically involves row reducing A to row echelon form or reduced row echelon form, solving Ax = 0, identifying the free variables, and expressing the general solution as a linear combination of vectors that correspond to these free variables. Each such vector is multiplied by an arbitrary parameter, and the collection of these coefficient vectors forms a basis for the kernel. Any basis of the kernel yields the same subspace, though the specific vectors may differ.

The rank-nullity theorem connects the kernel to the column space: rank(A) plus nullity(A) equals n, the number of columns. If the kernel is {0}, the nullity is zero and the kernbasis is an empty set.

Example: For A = [[1, 2, 3], [4, 5, 6]], the reduced row echelon form is [[1, 0, -1], [0, 1, 2]]. The free variable is x3, and setting x3 = t gives x = t[1, -2, 1]. Thus the kernbasis is { (1, -2, 1) }. Kernbasis is a fundamental tool for solving homogeneous systems and for understanding the structure of linear mappings.