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Linear Algebra Proof Question

  • Thread starter Zoe-b
  • Start date
  • #1
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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 Im (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?
 

Answers and Replies

  • #2
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So, what does "onto" mean?
 
  • #3
98
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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 Im T = W

?
 
  • #4
22,097
3,278
Edit: Is it correct just to write
T(V) = W since T is onto
so Im T = W

?
Yes, this is correct, but did you understand what you wrote?
 
  • #5
98
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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..
 
  • #6
22,097
3,278
Hmm, strange, there is nothing wrong with writing T(V)=W... So I have no idea why your tutor dislikes that...
 
  • #7
98
0
Ok, well as long as its considered right in general :P Thanks!
 

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