Is the transformation t' from V/ker(t) to W injective?

[iamalexalright]

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
$$V/ker(t) \approx im(t)$$

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 I am 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

 

[Nino MMP]

.I was thinking that maybe im(t) = V/ker(t) and then t' maps from V/ker(t) to W which is injective.But I don't really know how to prove this.Can someone please help me?Thanks in advance!