Discuss the following assertion:if two rational vector spaces have the same cardinal number(i.e., if there is some one-to-one correspondence between them), then they are isomorphic(i.e., there is linearity-preserving one-to-one correspondence between them).
Are two rational vector spaces having the same cardinal number are isomorphic?
-
0@ArturoMagidin Thanks very much for the tip my goal was n't to be rude. I mearly copied the question as was in the book. I would try rephrase in future with this in mind. – 2012-02-06
2 Answers
In general assuming the axiom of choice then there is a unique cardinality for all maximal linearly independent sets, i.e. bases.
This means that the idea of "dimension" is well defined, and two vector spaces whose dimension is the same are isomorphic. Infinities are quite varied and there are more than one size of infinity.
So there can be two vector spaces over a field $F$, both have an infinite dimension, but alas they are not isomorphic.
However, do note that any nonzero vector space contains a copy of the field, $F$. If this $F$ is infinite (such examples are $\mathbb Q$, $\mathbb R$, and other fields) then the vector space is infinite, as a set. The dimension, however, may still be finite.
-
0thanks for the clarification actually i was aware of Q and \mathbb RR not having the same cardinality fact, but not sure if i have come across yet what they are or how they are represented in mathematical literature. – 2012-02-06
No. Any two finite-dimensional vector spaces over $\mathbb{Q}$ have the same cardinality, but two such vector spaces are isomorphic if and only if they have the same dimension (which in general they don't).
-
2@Hardy: $\mathbb{Q}^n$ for every positive integer $n$. – 2012-02-06