(a) Let $X$ be an affine variety, $k[X]$ its ring of functions, and let $p \in X$. Assume $k$ is algebraically closed. Define the tangent space $T_{p} X$ at $p$. Prove the following assertions.

(i) A morphism of affine varieties $f: X \rightarrow Y$ induces a linear map

$$d f: T_{p} X \rightarrow T_{f(p)} Y$$

(ii) If $g \in k[X]$ and $U:=\{x \in X \mid g(x) \neq 0\}$, then $U$ has the natural structure of an affine variety, and the natural morphism of $U$ into $X$ induces an isomorphism $T_{p} U \rightarrow T_{p} X$ for all $p \in U$.

(iii) For all $s \geqslant 0$, the subset $\left\{x \in X \mid \operatorname{dim} T_{x} X \geqslant s\right\}$ is a Zariski-closed subvariety of $X$.

(b) Show that the set of nilpotent $2 \times 2$ matrices

$$X=\left\{x \in \operatorname{Mat}_{2}(k) \mid x^{2}=0\right\}$$

may be realised as an affine surface in $\mathbf{A}^{3}$, and determine its tangent space at all points $x \in X$.

Define what it means for two varieties $Y_{1}$ and $Y_{2}$ to be birationally equivalent, and show that the variety $X$ of nilpotent $2 \times 2$ matrices is birationally equivalent to $\mathbf{A}^{2}$.