Exercise 111.35.11. Let $k$ be an algebraically closed field. Let

be a morphism of schemes over $k$. This is given by $m$ polynomials $f_1, \ldots , f_ m$ in $n$ variables. Consider the matrix

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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