Matrix associated of an application between tangent spaces Let $M$ be a differential manifold and $X$ a vector field over $M$ s.t. $X(p) = 0$ for some $p \in M$. Let be $\phi_p : T_p(M) \to T_p(M)$ given as $$\phi_p(v) := [Y,X](p),$$ being $Y$ another vector field of $M$ s.t. $Y(p) = v$.
For $X = \sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ calculate the associated matrix of $\phi_p$ with respect the basis $\{(\frac{\partial}{\partial x_i})\}_{1 \leq i \leq m}$.
Can you help me, please?
And another question: Since $\phi_p(v) = [Y,X](p) = Y(X(p)) - X(Y(p)) = Y(0) - X(v) = -X(v)$, can we assert that $\phi_p$ does not depend of $Y$?
Thanks
 A: Your last computation is wrong (it actually doesn't make any sense to write $Y(X(p))$) and leads me to believe that the definition of $[Y,X]$ is not clear to you. Here is how it goes. Basically, you need to test how vector fields act on smooth functions. Let $f\in C^\infty(M)$ and let $X=X^i\partial_i$, $Y=Y^j\partial_j$, where by $\partial_i$ I denote $\partial/\partial_{x_i}$ and I use Einstein summation convention for conciseness. Then
$$
\begin{split}
[Y,X](f) &= Y(X(f))-X(Y(f)) \\
&= Y^j\partial_j(X^i\partial_if)-X^i\partial_i(Y^j\partial_jf) \\
&= Y^j(\partial_jX^i)\partial_if-X^i(\partial_iY^j)\partial_jf ,
\end{split}
$$
because the terms involving second order derivatives of $f$ cancel by Schwarz's theorem.
At point $p$, we have $X_p^i=0$ and $Y_p^j=v^j$, therefore only the first term remains
$$
[Y,X]_p = v^j(\partial_jX^i)_p\partial_i .
$$
In particular, this shows that $[Y,X]_p$ only depends on the value of $Y$ at point $p$, which is $Y_p=v$, and not on the actual vector field itself.
The matrix representation can be recovered from $\phi_p(v) = \phi_p(v)^i\partial_i$ from which
$$
\phi_p(v)^i = (\partial_jX^i)_p v^j = M^i_j v^j
$$
with $M^i_j=(\partial_jX^i)_p$. The transformation $\phi_p$ is linear and depends only on the derivatives of $X$ at $p$.
