One-to-one. Definition of kernel: Let be a linear transformation. The kernel of L,, is the




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Kernel and Range
Definition of one-to-one transformation:

A linear transformation is said to be one-to-one if for all in V, implies that (or implies ).
Example:
. Is L one-to-one?
[solution:]
Let . Then,



.

.

Therefore, L is one-to-one.


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 one-to-one if and only if


[proof:]
Since L is one-to-one, implies there is only one vector 0 such that .
. Suppose . Then,

. Since , that implies

. Therefore, L is one-to-one.
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 ,

.


  1. Is L onto?

  2. What is range L?

[solution:]




  1. 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 on-to.


(b)


Note:

As the linear transformation defined by



then


and


.
Important result:

Let be a linear transformation of an n-dimensional 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










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