Let $k$ be a field, let $X\subset \mathbb A_k^n$ be a closed subscheme, and let $p=(a_1,\dots,a_n)\in X(k)$. Let $S:=k[x_1,\dots,x_n]$, let $\mathfrak q_p:=(x_1-a_1,\dots,x_n-a_n)\trianglelefteq S$, and let $I_{X,p}\trianglelefteq S_{\mathfrak q_p}=\mathcal O_{\mathbb A_k^n,p}$ be the localized ideal defining $X$ at $p$. Suppose that $I_{X,p}$ is generated by the germs of polynomials $f_1,\dots,f_c\in S$, and suppose that $X$ has codimension $c$ at $p$, meaning equivalently that $\operatorname{height}(I_{X,p})=c$ in $\mathcal O_{\mathbb A_k^n,p}$. Define the [Jacobian matrix](/page/Jacobian%20Matrix) $J_p\in k^{c\times n}$ by

\begin{align*} (J_p)_{ij}:=\frac{\partial f_i}{\partial x_j}(p). \end{align*}

Then the Zariski tangent space satisfies

\begin{align*} T_pX=\ker(J_p:k^n\to k^c). \end{align*}

Consequently,

\begin{align*} \dim_k T_pX=n-\operatorname{rank}J_p. \end{align*}

In particular, $p$ is a smooth point of $X$ over $k$ if and only if $\operatorname{rank}J_p=c$; in that case,

\begin{align*} \dim_k T_pX=n-c. \end{align*}