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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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  • $\begingroup$ Your question asks 'one-to-one' and your text asks 'onto'. These are two separate things. Which do you want to know? $\endgroup$
    – Ian Coley
    Mar 22, 2013 at 2:56
  • $\begingroup$ Sorry, edited, I meant onto. $\endgroup$
    – Goose
    Mar 22, 2013 at 2:58
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    $\begingroup$ Well for a map to be onto, the dimension of the codomain (range) cannot exceed the dimension of the domain. Does that help? $\endgroup$
    – Ian Coley
    Mar 22, 2013 at 3:00
  • $\begingroup$ 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. $\endgroup$
    – Goose
    Mar 22, 2013 at 3:05
  • $\begingroup$ What is $F(R)$? $\endgroup$ Mar 22, 2013 at 3:11

1 Answer 1

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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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  • $\begingroup$ Make that $\gt$ into $\ge$, and I'll buy it. $\endgroup$ Mar 22, 2013 at 6:47
  • $\begingroup$ Ok, done. It's $0.05 Klingonian dollars, please...and thanks, of course. $\endgroup$
    – DonAntonio
    Mar 22, 2013 at 6:50
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    $\begingroup$ Sorry, all I have is upvotes. $\endgroup$ Mar 22, 2013 at 6:58
  • $\begingroup$ 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. $\endgroup$
    – Goose
    Mar 22, 2013 at 15:26
  • $\begingroup$ 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.. $\endgroup$
    – Goose
    Mar 22, 2013 at 21:46

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