some facts about injective and surjective linear maps
Let be a field and be vector spaces over .
We will show that .
Let . Then
where (note that the indexing set is ). Thus we have
It is clear that the equality implies that is surjective.
Proposition. Let be a surjective linear map. Then there exists a (injective) linear map such that .
Proof. Let be a basis of . Since is onto, then for any there exist such that . Now define by the formula
It is clear that , which implies that is injective.
If we combine these two propositions, we have the following corollary:
Corollary. There exists an injective linear map if and only if there exists a surjective linear map .
|Title||some facts about injective and surjective linear maps|
|Date of creation||2013-03-22 18:32:22|
|Last modified on||2013-03-22 18:32:22|
|Last modified by||joking (16130)|