I'm reading a book on differential forms and on page one it defines the tangent space to $\Bbb R^n$. In what follows I've translated the statements into two dimensions for simplicity.
Let $p$ be a point of $\Bbb R^2$. The set of vectors $q-p$, $q\in\Bbb R^3$ (that have origin an $p$) will be called the tangent space of $\Bbb R^2$ at $p$ and will be denoted by $\Bbb R^2_p$. The vectors $e_1=(1,0)$, $e_2=(0,1)$ will be identified with their translates $(e_1)_p$, $(e_2)_p$ at the point $p$.
So I get that $p$ is now the origin because $p-p=(0,0)$. But what happens to $e_1$ and $e_2$? Suppose $p=(1,1)$. Then $(e_1)_p=(0,1)-(1,1)=(0,-1)$ right? But that points down. Shouldn't $e_1$ translate to something that points in the same direction as $e_1$? My intuition tells me $(e_1)_p$ should point to $(1,2)$ which is the translation of $e_1$ to start at $(1,1)$.
So obviously my intuition is whack on this. What am I missing here and what's the right way to think about this? Any advice would be greatly appreciated. Thank you.
The translate $(e_1)_p$ is the unique vector satisfying $(e_1)_p - p = e_1$, so $(e_1)_p = p+e_1$. And, of course, similarly for $e_2$.
In your example $p = (1, 1)$, $(e_1)_p = (1,1)+(1, 0) = (2, 1)$ and $(e_2)_p = (1, 2)$.