Hi! this is not actually a homework problem, just a problem from a linear algebra book in spanish ( Algebra lineal, Elon Lages Lima) that has been bothering me A LOT lately because the "hint" says "it's actually much easier than it looks" :S any help would be appreciated a lot
1. The problem statement, all variables and given/known data
Let E and F be vector spaces (of any dimension) and let X be a subset of F with the following property (which is only a conditional):
IF a given linear transformation
A: E ---> F satisfies that X is a subset of Im(A)
THEN this linear transformation A is surjective. ... (*)
Prove that X is a generating set for F.
2. Relevant equations
E1)Im(A) = set of images of elements of E under A.
E2)A is surjective if and only if it transforms generating sets into generating sets.
E3)A has (at least one) right inverse if it is surjective
3. The attempt at a solution
I assumed the antecedent of (*) since I think I can start the proof that way. Then I can assert that on one hand that A is surjective and using E2) I concluded that Im(A) is a generating set which is redundant.
I also tried to take all y in F that are in Im(A) but not in X (because I assumed that X is inside Im(A) ) and get the inverse image of these elements so I can map these into X. But then I realized that I don't have the freedom to do that (I can at most assign arbitrary values to members of a basis).
I tried to use the contrapositive of (*) and assume that there exists some w in F such that it is not the image of any v in E. then it follows that there exists some x in X such that it is not the image of any v in E. But this got me nowhere.
Please help with this problem. Thanks