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 mia6 November 10th, 2010 08:02 AM

Isomorphism

This is a true/false question: All linear transformations from P3 to R2*2 are isomorphisms.
I think it is true because dim(P3)=dim(R2*2)=4, and ker(P3)=0. But the answer was false. I don't know why...So that means kernel is not 0?? But P3 has to be zero in order for R2*2 to be 0....
Thanks.

 Fernando Revilla November 10th, 2010 09:31 AM

Re: Isomorphism

It is false. If $E,\;F$ are vector spaces with $E\neq \left\{{0}\right\}$ , then

$0\;:\;E\rightarrow{F},\quad 0(x)=0\; \forall{x}\in{E}$

is a linear transformation but it is not injective.

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