Let $k$ be an algebraically closed field and $n \geqslant 1$. Exhibit $G L(n, k)$ as an open subset of affine space $\mathbb{A}_{k}^{n^{2}}$. Deduce that $G L(n, k)$ is smooth. Prove that it is also irreducible.

Prove that $G L(n, k)$ is isomorphic to a closed subvariety in an affine space.

Show that the matrix multiplication map

$$G L(n, k) \times G L(n, k) \rightarrow G L(n, k)$$

that sends a pair of matrices to their product is a morphism.

Prove that any morphism from $\mathbb{A}_{k}^{n}$ to $\mathbb{A}_{k}^{1}>\{0\}$ is constant.

Prove that for $n \geqslant 2$ any morphism from $\mathbb{P}_{k}^{n}$ to $\mathbb{P}_{k}^{1}$ is constant.