Suppose that are inverses.
If { } is a basis for a subspace of and , prove that { } is a basis for .
In addition, give an example to show that this need not be true if T does not have an inverse.
Take any transformation that is not invertible; for example, defined by . Since T is not invertible, . In fact, .
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.
"ker(T)" is the "kernel" of T, also called the "null space" of T. It is the subspace of all vectors, v, such that Tv= 0.
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.