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Cardinality and Dimention 
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#1
Nov1912, 05:55 PM

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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! 


#2
Nov1912, 07:20 PM

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#3
Nov2112, 07:41 PM

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