How do we know the Christoffel symbols can be defined this way? I am reading this article and they say one definition of the Christoffel symbols is
$$\partial_j e_i = \Gamma_{ij}^k e_k.$$
Here $(e_i)$ denotes the basis vectors for a coordinate system, and this is saying the (covariant) derivative of a basis vector with respect to coordinate $x^j$ can be equal to a linear combination of the same basis vectors with constant terms called Christoffel symbols.
I am just wondering, you can't actually do this in every setting right? I mean the derivative of an object lives in the tangent space and that is an object in a different space, how do we know it can be written as a linear combination of the original basis vectors? (In local coordinates -> flat space -> Euclidean space -> regular differentiation, nothing strange occurs because it follows calculus differentiation.)
 A: It sounds as though you may be getting a little confused by the fact that in Euclidean / Minkowskian space, one can think of "position vectors" defining points in the differentiable manifold itself.
But in the general setting, the notion of "vector" and "tensor" are only be defined in the tangent spaces to a given point. The manifold's points themselves cannot be thought of as vectors.
In the general setting, the tangent space to a point $p\in M$ in a differentiable manifold $M$ can be thought of as the space of derivatives $\mathrm{d}_\lambda \phi(\sigma(\lambda))|_{\lambda=0}$ of scalar functions $\phi:M\to\mathbb{R}$ of the points along $C^1$ paths $\sigma:[-1,\,1]\to M$ through that point (i.e. the path $\sigma$ passes through the point $p$ at $\lambda=0$). This space is a vector or linear space in the mathematical sense of the word: i.e. as decomposable into a linear superposition of basis vectors $e_i = \mathrm{d}_\lambda \phi(\zeta_i(\lambda))|_{\lambda=0}$.
Now, when we discuss the notion of a covariant derivative and the associated notion of parallel transport this latter notion is, by definition, the notion of a map between tangent spaces to neighboring points in the manifold that is linear. When the notion is incorporated into the covariant derivative the latter object must fulfill Leibnitz's rule. 
So the formula you cite is simply an expression of the linearity axiom of the definition: the image of the superposition of basis vectors is the superposition of the images of the individual basis vectors, i.e. what your formula is telling you.
