Then, is spanned by , however which is not a base.
This will work for any transformation that is not invertible:
Since it is not invertible, but for any , , and therefore the image of any base of will be mapped to the zero vector.
If T is invertible, then Tv= 0 gives or v= 0. That is, if T is invertible, its kernel (null space) consists only of the 0 vector.
In fact, you can also prove the other way: if the 0 vector is the only vector in the kernel of T, T is invertible.
"null space" is used exclusively in linear algebra. "kernel" of an operator is also used in group theory, ring theory, etc.