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!