Proving Onto-ness of Linear Transformations

[Zoe-b]

Homework Statement

I have this theorem in my notes but no proof and can't work out how to prove it (or find a proof via google):

Let T: V -> W be a linear transformation. Then T is onto iff I am (T) = W.

Homework Equations

not sure.

The Attempt at a Solution

I've only used this to prove other stuff.. so ideas on where to start proving this would be good. I understand the general idea that for T to be onto the image of T must contain the whole of W.. is that it?

 

[micromass]

So, what does "onto" mean?

 

[Zoe-b]

It means for all w in W there exists v in V st T(v) = w
.. am I being really slow?

Edit: Is it correct just to write
T(V) = W since T is onto
so I am T = W

?

 

[micromass]

Edit: Is it correct just to write
T(V) = W since T is onto
so I am T = W

?

Yes, this is correct, but did you understand what you wrote?

 

[Zoe-b]

yeah I do, only I don't think my tutor liked the notation T(V) = W when I used it last time. But maybe I used it wrong then..

 

[micromass]

Hmm, strange, there is nothing wrong with writing T(V)=W... So I have no idea why your tutor dislikes that...

 

[Zoe-b]

Ok, well as long as its considered right in general :P Thanks!