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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!

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# Cardinality and Dimention

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