Exercise 111.35.11. Let $k$ be an algebraically closed field. Let
\begin{eqnarray*} f : \mathbf{A}_ k^ n & \longrightarrow & \mathbf{A}^ m_ k \\ (x_1, \ldots , x_ n) & \longmapsto & (f_1(x_ i), \ldots , f_ m(x_ i)) \end{eqnarray*}
be a morphism of schemes over $k$. This is given by $m$ polynomials $f_1, \ldots , f_ m$ in $n$ variables. Consider the matrix
\[ A = \left( \frac{\partial f_ j}{\partial x_ i} \right) \]
Let $x \in \mathbf{A}^ n_ k$ be a closed point. Set $y = f(x)$. Show that the map on tangent spaces $T_{\mathbf{A}^ n_ k/k, x} \to T_{\mathbf{A}^ m_ k/k, y}$ is given by the value of the matrix $A$ at the point $x$.
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