# Cardinality and Dimention

by rhobymic
Tags: cardinality, dimention
 P: 5 This is not a homework question ..... If two vector spaces, say V and W, have equal cardinality |V|=|W| .... do they then have the same dimension? That is dim(V)=dim(W)? I am struggling with making this call one way or the other. This is no area of expertise for me by any means so I know I am missing something important but here are my thoughts: -> No it does not mean they have the same dim. Dimension is the value of the cardinality of the BASIS vectors of a vector space not the cardinality of the full vector space. -> Yes it does because if |V|=|W| is true then there is a bijection between V and W and therefor an isomorphic linear transformation T between V and W. This would imply that T carries a basis from V into W and so V and W would have the same cardinality of basis vectors er go the same dimension.... I am still leaning towards "No" because I think the assumption that if V and W are bijective then there is an isomorphic linear transformation is probably not possible... Thanks for any help!
Emeritus
Thanks
PF Gold
P: 15,867
 Quote by rhobymic This is not a homework question ..... If two vector spaces, say V and W, have equal cardinality |V|=|W| .... do they then have the same dimension? That is dim(V)=dim(W)? I am struggling with making this call one way or the other. This is no area of expertise for me by any means so I know I am missing something important but here are my thoughts: -> No it does not mean they have the same dim. Dimension is the value of the cardinality of the BASIS vectors of a vector space not the cardinality of the full vector space.
This is a nice observation, but it is not a proof. To answer the question as "no", you just need to come up with two vector spaces that have equal cardinality but not equal basis.

 -> Yes it does because if |V|=|W| is true then there is a bijection between V and W and therefor an isomorphic linear transformation T between V and W. This would imply that T carries a basis from V into W and so V and W would have the same cardinality of basis vectors er go the same dimension....
I don't really see why an arbitrary bijection would be an isomorphism...
P: 297
 Quote by rhobymic -> Yes it does because if |V|=|W| is true then there is a bijection between V and W and therefor an isomorphic linear transformation T between V and W. This would imply that T carries a basis from V into W and so V and W would have the same cardinality of basis vectors er go the same dimension.... Thanks for any help!
A bijection needs not be linear. In fact, Cantor proved that R and R2 have the same cardinality, and hence that all Rn (n>0) have the same cardinality. For a proof, see Theorem 2 here. (The proof is actually incomplete, since the author has forgotten the 0.99999.....=1 problem. But it can be fixed.)

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