Image and kernel

When dealing with linear mappings, we will often encounter two important terms: the image and the kernel, both of which are vector subspaces with rather important properties.

The kernel (sometimes called the null space) is 0 (the zero vector) and is produced by a linear map, as follows:

And the image (sometimes called the range) of T is defined as follows:

such that .

V and W are also sometimes known as the domain and codomain of T. 

It is best to think of the kernel as a linear mapping that maps the vectors  to . The image, however, is the set of all possible linear combinations of  that can be mapped to the set of vectors . 

The Rank-Nullity theorem (sometimes referred to as the fundamental theorem of linear mappings) states that given two vector spaces V and W and a linear mapping , the following will remain true:

.