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How are R^2 and R^4 isomorphic as VS's over Q?by joeblow
Tags: tensor product 
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#1
Mar2411, 08:56 PM

P: 71

I came across a problem asserting that the Ctensor product of C and C and the Rtensor product of C and C are isomorphic as Qmodules. How does this begin to make sense as they do not have the same dimension over R? The first is isomorphic to R^{2} and the second is isomorphic to R^{4} over R.
Also, I'm struggling with tensors. Does anyone have a good source I should check out? (I've tried many) 


#2
Mar2511, 04:59 AM

Sci Advisor
P: 1,716

The tensor product is a way to extend the field of scalars of a vector space. A real vector space of dimension n (n can be infinite) becomes a complex vector space of dimension n when tensored with C. It becomes a real vector space of twice the dimension over R. If one views R as a 1 dimensional vector space over itself with single basis vector v, the all elements are of the form rV for real numbers,r. Over C the basis is also v and the resulting vector space is 1 dimensional over C. But over R the basis is V and iV and so is 2 dimensional over R. 


#3
Mar2511, 03:55 PM

Sci Advisor
P: 905




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