Kernel and Range
Definition of onetoone transformation:
A linear transformation is said to be onetoone if for all in V, implies that (or implies ).
Example:
. Is L onetoone?
[solution:]
Let . Then,
.
.
Therefore, L is onetoone.
Definition of kernel:
Let be a linear transformation. The kernel of L, , is the subset of V consisting of all vectors such that .
Example:
Let
.
What is ?
[solution:]
is the set consisting of all vectors such that
.
That is, is the solution space of .
Important result:
Let be a linear transformation. Then, is a subspace of V.
[proof:]
For any , . Then,
1. .
2. .
By 1, 2, is a subspace of V.
Important result:
A linear transformation is onetoone if and only if
[proof:]
Since L is onetoone, implies there is only one vector 0 such that .
. Suppose . Then,
. Since , that implies
. Therefore, L is onetoone.
Important result:
Let be a linear transformation defined by , where A is a matrix. Then, the following conditions are equivalent:
Definition of range:
Let be a linear transformation. The range of L, , is the subset of W consisting of all vectors that are images of vectors in V (i.e.,
for any , there exists in V such that ).
If , L is said to be onto.
Important result:
Let be a linear transformation. Then, is a subspace of V.
[proof:]
Since , . Thus, is not a empty set. For any , there exist in V such that . Then,
1. . Then, since .
2. . Then, since
By 1, 2, is a subspace of V.
Example:
Let ,
.

Is L onto?

What is range L?
[solution:]

Suppose L is onto. Then for any vector , there exists such that
.
That is span is a basis for .
But
.
Thus,
is not a basis and it is a contradiction. Therefore, L is not onto.
(b)
Note:
As the linear transformation defined by
then
and
.
Important result:
Let be a linear transformation of an ndimensional vector space V into a vector space W, then,
,
where is the dimension of some vector space.
Note:
As the linear transformation defined by
the above important result is
