How are R^2 and R^4 isomorphic as VS's over Q?


by joeblow
Tags: tensor product
joeblow
joeblow is offline
#1
Mar24-11, 08:56 PM
P: 71
I came across a problem asserting that the C-tensor product of C and C and the R-tensor product of C and C are isomorphic as Q-modules. How does this begin to make sense as they do not have the same dimension over R? The first is isomorphic to R2 and the second is isomorphic to R4 over R.

Also, I'm struggling with tensors. Does anyone have a good source I should check out? (I've tried many)
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
lavinia
lavinia is offline
#2
Mar25-11, 04:59 AM
Sci Advisor
P: 1,716
Quote Quote by joeblow View Post
I came across a problem asserting that the C-tensor product of C and C and the R-tensor product of C and C are isomorphic as Q-modules. How does this begin to make sense as they do not have the same dimension over R? The first is isomorphic to R2 and the second is isomorphic to R4 over R.

Also, I'm struggling with tensors. Does anyone have a good source I should check out? (I've tried many)
Over Q these vector spaces are infinite dimensional. They will be isomorphic if the cardinality of their bases is the same - which I think is the cardinality of the Continuum.

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.
Landau
Landau is offline
#3
Mar25-11, 03:55 PM
Sci Advisor
P: 905
Quote Quote by lavinia View Post
Over Q these vector spaces are infinite dimensional. They will be isomorphic if the cardinality of their bases is the same - which I think is the cardinality of the Continuum.
Indeed; we discussed this here.


Register to reply

Related Discussions
isomorphic Calculus & Beyond Homework 12
when is GAl(E\Q) isomorphic to Zn Linear & Abstract Algebra 0
Isomorphic Help General Math 1
Z and Q are not isomorphic Calculus & Beyond Homework 4
isomorphic Calculus & Beyond Homework 7