Let $\nabla$ be a connection on the tangent bundle $T\mathbb{R}^n$. Now, I need to show that there exist smooth function $C_i: \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$, $i=1,\dots ,n$ such that
$$\nabla_{\frac{\partial}{\partial x^i}} = \frac{\partial}{\partial x^i} + C_i$$
for all $i$, where $ \frac{\partial}{\partial x^i}$ are the coordinate vector fields.
Can someone give me hints how to solve this. I also don't quite understand what is meant by this formula as this $C_i$ is a matrix, whereas $ \frac{\partial}{\partial x^i}$ is a vector field and $\nabla_{\frac{\partial}{\partial x^i}} $ is a map taking a vector field and giving a vector field.
Attempt: I computed that the j-th row of the matrix $C_i(p)$ is given by $(\nabla_{\frac{\partial}{\partial x^i}})(\frac{\partial}{\partial x_j})(p)$. Is that true?