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Is it possible for a linear map to be onto if:

  1. The domain is $R^5$ and the range is $R^4$?
  2. The domain is $R^5$ and the range is $M(4,4)$?
  3. The domain is $R^5$ and the range is $F(R)$?

I know to be onto $\operatorname{rank} T = \dim W$ for $T\colon V \to W$.

My thoughts:

  1. No, because $5 + \operatorname{nullity}T = 4$, thus $\operatorname{nullity}T = -1$ which isn't possible?
  2. No, because 5 and $\dim M = 8$ are not equal?
  3. Not sure about this one.

Any help would be appreciated, thanks.

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Your question asks 'one-to-one' and your text asks 'onto'. These are two separate things. Which do you want to know? –  Ian Coley Mar 22 '13 at 2:56
    
Sorry, edited, I meant onto. –  Goose Mar 22 '13 at 2:58
1  
Well for a map to be onto, the dimension of the codomain (range) cannot exceed the dimension of the domain. Does that help? –  Ian Coley Mar 22 '13 at 3:00
    
Ahh, so 1. is Yes? Is the dimension of range of M(4,4) = 8? And as for 3. I'm not sure what the dimension of F(R) is. –  Goose Mar 22 '13 at 3:05
    
What is $F(R)$? –  Avi Steiner Mar 22 '13 at 3:11

1 Answer 1

I think you may want to concentrate in one single lemma: let $\,V\,,\,W\,$ be two linear spaces over the same field, then

Claim: There exists an onto linear map $\,V\to W\,$ iff $\,\dim V\ge\dim W\,$

The above is true even if the dimensions are infinite (at least with the aid of AC).

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Make that $\gt$ into $\ge$, and I'll buy it. –  Gerry Myerson Mar 22 '13 at 6:47
    
Ok, done. It's $0.05 Klingonian dollars, please...and thanks, of course. –  DonAntonio Mar 22 '13 at 6:50
1  
Sorry, all I have is upvotes. –  Gerry Myerson Mar 22 '13 at 6:58
    
I understand that seeing as how Frank posted something similar above: "Well for a map to be onto, the dimension of the codomain (range) cannot exceed the dimension of the domain." I've figured out the first two, but I can't figure out the dimension of F(R). If you could help me out on that I'll accept your answer. –  Goose Mar 22 '13 at 15:26
    
I apologize if I sounded rude, that was not my intent. As far as what F(R) is, this problem I posted comes straight from my book, I suppose it assumes you should already know what F(R) is like you know what R4 and P5 are.. –  Goose Mar 22 '13 at 21:46

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