Prove Corollary of Rank-Nullity theorem

[iamalexalright]

Homework Statement

Prove:
Let $$\tau \in L(V,W)$$, where dim(V) = dim(W) < infinity. Then $$\tau$$ is injective iff it is surjective.

Homework Equations

L(V,W) is the set of all linear transformations from V to W.

1. Any complement of ker(t) is isomorphic to im(t)
2. dim(ker(t)) + dim(im(t)) = dim(V)

The Attempt at a Solution

I'm pretty lost in starting this.
I know it is surjective iff im(t) = W
I know it is injective iff ker(t) = {0}

Should I assume its injective but not surjective (to move towards a contradiction)?

And maybe I don't understand the concept of an isomorphism but if:
$$im(\tau) = W$$ and
$$ker(\tau)^{c} \approx im(\tau)$$
then how does the ker(t)^c relate to W?

 

[micromass]

You can prove this by only looking at the formula dim(ker(t))+dim(im(t))=dim(V).

Assume that t is injective. As you stated, this implies that ker(t)={0}. Thus dim(ker(t))=0.
So, what does this imply in the above formula?

 

[iamalexalright]

that implies dim(im(t)) = dim(V) = dim(W)

that doesn't necessarily imply though that im(t) = W

am I just not thinking about it enough?

 

[micromass]

That's correct. But since $$im(t)\subseteq W$$ and dim(im(t))=dim(W) (as proven), then this DOES imply that im(t)=W.

In general, if you have two finite-dimensional spaces V and W such that $$V\subseteq W$$ and dim(V)=dim(W), then V=W!

 

[iamalexalright]

Ah ! okay, that makes perfect sense

thanks for the help