Assume DC($\aleph_1$).

Can we define the following:

1. Basis for a vector space $V$ over a field $K$ such that $\operatorname{card}(K) \leq \aleph_1$ and we happen to find a generating set of $V$ of cardinality $\leq \aleph_1$.
2. Linear dimension making the same assumption as above.
3. Transcendence degree of a field $K$ over $\mathbb{Q}$ if we happen to know that $K$ is of cardinality $\leq \aleph_1$. etc...