I'm reading An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised by Boothby.
I'm reading the second chapter right now, and in this chapter, the tangent space $T_p(\mathbb{R}^n)$ for $p \in \mathbb{R}^n$ is defined by the vector space of line segment starting at $p$.
The canonical basis, $E_{1a}, \ldots, E_{na}$ for $T_a(\mathbb{R}^n)$ is the line segment connecting $a$ to $a + e_i$ where $e_i = (0,0,....,1,0...0)$.
Now, the vector field on $U \subseteq \mathbb{R}^n$ is defined as a function sending $p \in U$ to a vector $X_p \in T_p(U)$.
Now it says to define vector field $E_i$ by sending each $p \in \mathbb{R}^n$ to the naturally defined basis vector.
I assume this means that for each $p$, send it to $E_{ip} \in T_p(\mathbb{R}^n)$. So for example, the vector field $E_1$ would send $(1,1)$ to line segment connecting $(1,1)$ and $(2,1)$.
Now, they set $U = \mathbb{R}^2 - 0$ and define $X_1 = x_1 E_1 - x_2 E_2$ and $X_2 = -x_2 E_1 +x_1 E_2$.
So I assume this is the vector field sending $p = (x_1, x_2)$ to the vector $x_1 E_1 - x_2 E_2 \in T_p(\mathbb{R}^2)$.
Now here is the problem. Let's say $p = (1,1)$. Then $X_1$ sends $p$ to $E_{1p} - E_{2p} \in T_p(\mathbb{R}^2)$, which is a line segment connecting $(1,1)$ to $(2,0)$, which would go against what the figure says.
And $X_2$ would send $p$ to a line segment connecting $(1,1)$ to $(0,2)$, which is against the figure again.
So what am I misunderstanding here?
Thank you!