tangent spaces on two different points on a manifold If $p$ and $q$ are two points on a $n$-dimensional manifold $M$, then their tangent spaces are two $n$-dimensional vector spaces. So algebraically they have the same structure, but they are not the same because $p$ and $q$ are different points. I can understand they are not the same by visualization. How is it stated algebraically? 
I mean if M is 2 dimensional I can picture two different planes, one on $p$ and one on $q$. How is that difference stated mathematically?
Thanks.
 A: The general construction you're looking for is that of the tangent bundle (or, more generally, a vector or fiber bundle). The individual tangent spaces are isomorphic; they're vector spaces of the same dimension, and there's a canonical basepoint. The important point, though, is that they vary continuously. More specificially, each point $p\in M$ has a neighborhood $U$ with a homeomorphism $TU \to U \times \mathbb{R}^n$ such that the projection $TU \to U \times \mathbb{R}^n \to U$ is just the map sending the tangent space $T_p M$ to $p$. (Unravelling the definition of tangent space for a manifold will show where this map comes from.) It's not true that $TM = M\times \mathbb{R}^n$; the point is that the tangent bundle is (generally) locally trivial but not actually trivial, i.e., a product. Even in the particular case of tangent spaces to manifolds, it may be easier to think of them abstractly rather than as embedded in some large $\mathbb{R}^N$.
A: The algebraic statement is very simple: 
Even though the spaces $T_pM$ and $T_qM$ are isomorphic, there are many different isomorphisms between these two spaces, none of which is "special" in any way or can be distinguished from the others. This is why one cannot state that the two spaces are equal to one another.
This is a serious matter which leads to deep theories. One simple example for this difference between isomorphism and equality is the following: let $\gamma:[0,1]\to M$ be a path, and let $X$ be a vector field along $\gamma$. That is, $X(t)$ is a tangent vector at $\gamma(t)$ for every $t\in[0,1]$. Now, one can wonder if $X$ is constant. However, this depends on how one identifies the different tangent spaces $T_{\gamma(t)}M$ to one another. If $M$ doesn't have some extra structure (such as a connection on the tangent bundle), then the statement "$X$ is constant" is just meaningless.
