Kernel (linear algebra) - Wikipedia

Abstraction: Subspace of a linear map's domain mapped to the zero vector

Key points:

• Kernel (null space) of L: V→W is ker(L) = {v in V : L(v) = 0}; always a linear subspace of the domain
• Rank-nullity theorem: dim(ker(L)) + rank(L) = dim(V); nullity is the dimension of the kernel
• Kernel is the orthogonal complement of the row space; vectors in ker(A) are orthogonal to all row vectors of A
• Four fundamental subspaces of matrix A: kernel, row space, column space (image), and left null space (cokernel)
• Solution set to Ax = b is the translation of ker(A) by any particular solution v: {v + w : w in ker(A)}
• Computed via Gaussian elimination (column echelon form); for floating-point matrices, requires well-conditioned A

Connections: Linear Algebra · Null Space

Source: http://en.wikipedia.org/wiki/Kernel_(linear_algebra)