# linear algebra/ group theory question [closed]

Just one I'm not sure how to figure my way through:

$\mathbb{R}[X] \cong \mathbb{R}[X] \oplus \mathbb{R}[X]$

Thanks so much!

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Just a thought, does it help to think of $\mathbb{R}[X]=\langle X,X^3,X^5,\dots\rangle\oplus\langle 1,X^2,X^4,\dots\rangle$? –  yunone Nov 28 '12 at 5:51
Yes certainly. But why is $\mathbb{R}[X]$ isomorphic to that tensor product? –  Thomas Nov 28 '12 at 6:06
Tensor product? I assumed you were using $\oplus$ to denote the direct sum. If you did intend direct sum, the isomorphism should follow by the fact that two vector spaces over the same field are isomorphic if and only if they have the same dimension. This works for infinite dimensional spaces as well. –  yunone Nov 28 '12 at 6:10
Yeah sorry I meant direct sum. Up too late.. –  Thomas Nov 28 '12 at 6:14
What kind of isomorphism are you looking for? Of sets, groups, vector spaces, algebras? –  Phira Nov 28 '12 at 13:00