If A and B are vector spaces such that
A = B
Do they have the same dimension?
I think yes, because if they are equal then they can be spanned by some vectors. Any vectors that span A span B. Suppose then that A has a smaller dimension, meaning that it can be spanned by a smaller number of linearly independent vectors. Well, if these vectors span A, then they must be able to span B as well meaning that the dimensions are equal. Thus, A cannot have a smaller dimension (different dimension) than B.
Let me know if my logic is incorrect.