A linear map's kernel is everything it sends to zero
Every linear transformation has a blind spot: the collection of inputs it crushes down to zero. Mathematicians call this the kernel, or null space. Knowing its size tells you exactly how much information the map throws away, and two inputs end up at the same output precisely when their difference lies in that blind spot.
Take a linear map L from one vector space V to another. Its kernel consists of every vector v in V with L(v) equal to the zero vector. This set is never just a random scattering of points: it always contains zero, and adding two of its members or scaling one by any number keeps you inside it. In other words, it is itself a subspace of V.
The kernel measures lost information. If two vectors differ by something in the kernel, the map cannot tell them apart. That leads to the rank–nullity theorem for finite dimensions: the dimension of the kernel, called the nullity, plus the dimension of the image, called the rank, always equals the dimension of the starting space. Whatever dimensions the map does not preserve, it must collapse.
For a matrix A with m rows and n columns, the kernel is the solution set of the homogeneous system A times x equals zero. Each entry of A times x is the dot product of one row of A with x, so x lies in the kernel exactly when it is perpendicular to every row. The kernel is therefore the orthogonal complement of the row space, and rank plus nullity equals n, the number of columns.
The concept stretches well beyond ordinary matrices. It works for maps between modules, where scalars come from a ring instead of a field, though rank and nullity may no longer apply there. For topological vector spaces with a finite-dimensional target, a linear map is continuous exactly when its kernel is a closed subspace. There is also a mirror image, the left null space or cokernel, built from the transpose.
Source: Kernel (linear algebra)
