= Solution
For a <projective scheme> $X$ over $k$, define its <Hilbert functor> on $k$-schemes by assigning to $T$ the set of <closed subschemes> $W\subseteq X\times_kT$ such that $W\to T$ is a <flat morphism> of finite presentation and is proper. For projective $X$, the properness is automatic for these closed families. A morphism $T'\to T$ sends a family to $W\times_TT'\subseteq X\times_kT'$; <base change> preserves the required properties, making this a contravariant <functor>. The embedding in $X\times T$ is part of the data, so these are embedded families rather than families modulo automorphisms of $X$.
Choose a very ample <line bundle> on $X$. In a flat projective family of finite presentation the fibre <Hilbert polynomial> is locally constant. Requiring polynomial $P$ gives a subfunctor represented by $\operatorname{Hilb}^P(X)$; the unrestricted <Hilbert functor> is represented by the disjoint union over $P$. For the specified $Z$, use its polynomial $P_Z$ and let $h=[Z]$ be the resulting $k$-point of the <Hilbert scheme> $H$.
A <Zariski tangent space> vector at $h$ is a $k$-morphism
$$
v:\operatorname{Spec}D\longrightarrow H,\qquad D=k[\varepsilon]/(\varepsilon^2),
$$
whose restriction to $\operatorname{Spec}k$ is $h$. Here $D$ is the ring of <dual numbers>. Locally such a morphism sends $a$ to $a(h)+\varepsilon d(a)$, where $d(ab)=a(h)d(b)+b(h)d(a)$, so these morphisms form the vector space $\operatorname{Hom}_k(\mathfrak m_h/\mathfrak m_h^2,k)$. By the representing property of the <Hilbert scheme>, the same tangent vector is exactly a <first-order embedded deformation>
$$
Z_\varepsilon\subseteq X\times_k\operatorname{Spec}D,\qquad Z_\varepsilon\text{ flat over }D,\qquad Z_\varepsilon\times_Dk=Z
$$
with the last equality an equality of embedded <closed subschemes>.
Let $\mathcal I$ be the <ideal sheaf> of $i:Z\hookrightarrow X$, and define its <normal sheaf> by
$$
\mathcal N_{Z/X}=\mathcal Hom_{\mathcal O_Z}(\mathcal I/\mathcal I^2,\mathcal O_Z).
$$
This definition is valid even when $X$ or $Z$ is singular; it need not give a locally free sheaf. We now derive the tangent-space identification by constructing inverse maps, rather than merely invoking the normal sheaf's name.
Work on an <affine open subscheme> $\operatorname{Spec}A\subseteq X$, put $B=A/I$, and let $J\subseteq A\oplus\varepsilon A$ define a flat deformation with special fibre $B$. The <flatness criterion over dual numbers> says that a $D$-module $M$ is flat exactly when
$$
\ker(\varepsilon:M\to M)=\varepsilon M.
$$
Flatness implies this by tensoring $0\to(\varepsilon)\to D\to k\to0$. Conversely lift a $k$-basis of $M/\varepsilon M$ to $M$. It generates $M$ over $D$: after removing the linear combination representing an element modulo $\varepsilon$, the remainder is $\varepsilon$ times another element, whose residue is another finite basis combination. For independence, reduce a relation modulo $\varepsilon$ to remove its constant coefficients. The remaining relation says that a linear combination of the lifts belongs to the kernel of $\varepsilon$, hence to $\varepsilon M$, so its remaining coefficients vanish modulo $\varepsilon$ too. The lifts are a free $D$-basis, establishing flatness.
Applying this criterion to $M=(A\oplus\varepsilon A)/J$ gives
$$
J\cap\varepsilon A=\varepsilon I.
$$
Indeed $\varepsilon I\subseteq J$ follows by lifting each $f\in I$ to $f+\varepsilon g\in J$ and multiplying by $\varepsilon$. Conversely, if $\varepsilon a\in J$, then the class of $a$ is in the kernel of $\varepsilon$ on $M$, hence in $\varepsilon M$; reducing modulo $\varepsilon$ gives $a\in I$.
For $f\in I$, choose a lift $f+\varepsilon g\in J$ and define
$$
\phi(f)=g\bmod I\in B.
$$
Two lifts differ by an element of $J\cap\varepsilon A=\varepsilon I$, so this is well defined. Addition and multiplication of lifts by elements of $A$ prove that $\phi$ is $A$-linear. For $f_1,f_2\in I$, one has $\phi(f_1f_2)=f_1\phi(f_2)=0$ in $B$, so it factors through an element of $\operatorname{Hom}_B(I/I^2,B)$.
Conversely, for such a homomorphism define
$$
J_\phi=\{f+\varepsilon g:f\in I,\quad g\bmod I=\phi(f)\}\subseteq A\oplus\varepsilon A.
$$
This is an <ideal>: multiplying by $a+\varepsilon b$ gives $af+\varepsilon(ag+bf)$, and $ag+bf\bmod I=a\phi(f)=\phi(af)$. Its reduction is $I$. To prove that its quotient is flat, suppose $\varepsilon[a+\varepsilon b]=0$. The condition $\varepsilon a\in J_\phi$ says $a\in I$. Choose $g$ lifting $\phi(a)$; then $a+\varepsilon g\in J_\phi$, so
$$
[a+\varepsilon b]=\varepsilon[b-g].
$$
Thus the kernel of $\varepsilon$ is its image, and the criterion proves flatness. These constructions are inverse: the graph condition reconstructs every lift in $J$, and changes by $\varepsilon I$ account for all choices.
<Localization> respects both constructions. The local maps $I/I^2\to B$ therefore glue exactly to <global sections> of the <internal Hom sheaf> $\mathcal N_{Z/X}$; conversely such a section gives compatible local ideals that glue to $Z_\varepsilon$. Hence
$$
\boxed{T_{[Z]}\operatorname{Hilb}(X)\cong\operatorname{Hom}_{\mathcal O_X}(\mathcal I,i_*\mathcal O_Z)\cong H^0(Z,\mathcal N_{Z/X}).}
$$
The middle identification uses that every map to $\mathcal O_Z$ kills $\mathcal I^2$. The zero map gives the product deformation $Z\times\operatorname{Spec}D$; addition and scalar multiplication of the maps $\phi$ give the canonical vector-space operations on the tangent space. This proves the <Zariski tangent space of a Hilbert scheme> formula without a smoothness or regular-embedding hypothesis.
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