Help understand isomorphism of tensor product and connection to vector spaces well I'm having a hard time understanding the tensor product. Here's a problem from Atiyah and Macdonald's book:
Let $A$ be a non-trivial ring and let $m,n$ be positive integers. Let $f: A^{n} \rightarrow A^{m}$ be an isomorphism of $A$-modules. Show this implies that $n=m$. 
Well the solution is at follows:
Let $m$ be a maximal ideal of $A$. Then we have an induced isomorphism:
$(A/m) \otimes_{A} A^{n} \rightarrow (A/m) \otimes_{A} A^{m}$.
Now it says that this is an isomorphism between vector spaces of dimension $n$ and $m$.
My questions are:
1) How do we know that $(A/m) \otimes_{A} A^{n}$ is a vector space over $A/m$ ?
2) How do we know it has exactly dimension $n$?
Is there some "standard" theorem that tells us this? Can you please explain this in detail?
Thanks

1) Let $f: A \rightarrow B$ be any homomorphism of commutative rings and let $M$ be any $A$-module.  Then one may view $B \otimes_A M$ as a $B$-module via $b \cdot (\sum_i b_i \otimes x_i) := \sum_i b b_i \otimes x_i$.  This process of starting with an $A$-module and getting a $B$-module is called base change.
In your particular case the map is $f: A \rightarrow A/\mathfrak{m}$, so $(A/\mathfrak{m}) \otimes_A M$ is an $A/\mathfrak{m}$-module.  But since $\mathfrak{m}$ is maximal, $A/\mathfrak{m}$ is a field, and a module over a field is simply a vector space over that field.  This answers your first question.
2) You should establish the following two properties of tensor products:
(i) For any $A$-module $M$, there is a canonical isomorphism $M \otimes_A A \stackrel{\sim}{\rightarrow} M$.  (Thus "changing the base from $A$ to $A$" has no effect.)
(ii) For any family $M_i$ of $A$-modules indexed by a set $I$ and any $A$-module $M$, there is a canonical isomorphism $(\bigoplus_{i \in I} M_i) \otimes_A N \stackrel{\sim}{\rightarrow} \bigoplus_{i \in I} (M_i \otimes_A N)$.  
What you want follows easily from these facts (which are themselves ubiquitously useful).  Try it yourself and ask again if you have further questions.
