# Is it possible for these to be onto?

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?

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
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

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 $\,\dim V\ge\dim W\,$
Make that $\gt$ into $\ge$, and I'll buy it. – Gerry Myerson Mar 22 '13 at 6:47