Counting types in theory of vector spaces? I am working on these two problems:

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*For $T$ the theory of $\mathbb Q$-vector spaces, $\mathcal M\models T$, and $A\subseteq M$, show that $\lvert S_1^\mathcal{M}(A)\rvert\leq\lvert A\rvert+\omega$.


*For $T$ the theory of infinite $\mathbb Q$-vector spaces, $S_1(T)$ is finite but $S_2(T)$ is infinite.
We are using the language $\mathcal L=\langle +,0,(\lambda_a)_{a\in\mathbb Q}\rangle$ where $\lambda_a$ represents scalar multiplication by $a$.
For 1) If $\mathcal M$ is countable, then every $p\in S_1(A)$ is realized by some $x\in\mathbb Q^\omega$ (the countable saturated model), and since this is countable that means at most countably many types can be realized, so $\lvert S_1(A)\rvert\leq \omega=\lvert A\rvert+\omega$. But I am not sure what to do if $\mathcal M$ is uncountable.
For 2) I know that $S_n(T)$ is countable for all $n$, but I don't know how to distinguish whether it is finite or countably infinite for a given $n$. Is there a standard way to count 1-types, or types of a specific $n$ in general?
 A: "Is there a standard way to count $1$-types, or types of a specific $n$ in general?"
Yes, there are two basic principles that are useful for counting types:

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*If $a$ and $b$ satisfy all the same formulas over $C$, then they have the same complete type over $C$. This condition is rather tautological, because the complete type of a tuple over $C$ is, by definition, the set of all formulas over $C$ satisfied by the tuple. But the point is that if you can understand all formulas over $C$ up to equivalence, then you can often classify all complete types over $C$. Usually, the strategy is to prove a quantifier elimination result (possibly in some definable expansion of the language), and then classify types according to which atomic formulas they contain.


*If there is an automorphism fixing $C$ pointwise and moving $a$ to $b$, then $a$ and $b$ have the same complete type over $C$. The converse is true in a $|C|^+$-saturated model, but if we only want an upper bound on the number of types, we often don't need to explicitly invoke saturated models.
You can answer your questions using either principle. To use the first, you need to know that the theory of infinite $\mathbb{Q}$-vector spaces has quantifier elimination. To use the second, you need to know that vector spaces have lots of automorphisms: any two linearly independent tuples of the same length in a vector space $V$ are conjugate by an automorphism of $V$.
