Kernel extension theorem

The kernel extension theorem for vector spaces (also called, though it is a corollary of the Dimension theorem for vector spaces) is the following:

Given a Linear transformation T that maps from a certain domain to a certain codomain, the Hamel dimension of the domain is equal to the Hamel dimension of the transformation's range plus the Hamel dimension of the kernel.

In formula form:

Given T: U->V, dim(range(T)) + dim(kernel(T)) = dim(U)