Linear Map - Kernel, Image and The Rank–nullity Theorem

Kernel, Image and The Rank–nullity Theorem

If f : V → W is linear, we define the kernel and the image or range of f by

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is known as the rank–nullity theorem:

The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.