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arkanoid
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Homework Statement
Let [tex] A: E \rightarrow F [/tex] be a linear transformation between vector spaces (of any dimension) and let X be a subset of F with the following property (which is only a conditional):
IF [tex] X \subseteq Im(A) [/tex] THEN A is surjective. ... (*)
Prove that X is a generating set for F.
Homework 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 right inverse if it is surjective
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
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