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First Isomorphism Proof

  • #1

Homework Statement


Let t:V -> W be a linear transformation. Then the transformation t':V/ker(t) -> W defined by:
t'(v + ker(t)) = tv is injective
and
[tex]V/ker(t) \approx im(t)[/tex]


Homework Equations


A previous theorem:
Let S be a subspace of V and let t satisfy S <= ket(t). Then there is a unique linear transformation t':V/S -> W with the property that
t'*pi_s = t where pi_s is a transformation from V to V/S.

Morever, ker(t') = ker(t)/S and im(t') = im(t)



The Attempt at a Solution



The book says the first isomorphism theorem follows from the other theorem posted above.
First, do I need to prove that t' is injective (wondering because the theorem states it)?

Other than that Im kinda confused at where to go.

If t' is injective then ker(t') = ker(t) = {0}, right?
Then V/ker(t) = V... but this is all doesn't seem like the right way to go
 

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