Let $\mathbf{e_j}$ be a local coordinate frame on the manifold $M$.
How do I compute/obtain the coefficients $A_j^k$ so that $$\mathbf{e_j} (v^i) = A_j^k \partial_k (v^i)= A^k_j \frac{\partial v^i}{\partial x^k}$$.
In other words how do I obtain the real numbers $A_j^k$ so that $$\mathbf{e_j} = A_j^k \partial_k$$?
What is the formula for these numbers?