Why is $\mathbb{P}(Sym_2(\mathbb{C}^2))$ isomorphic to $\mathbb{P}^2(\mathbb{C})$?

Question

Let $Sym_2(\mathbb{C}^2)$ denote the space of symmetric 2-tensors on $\mathbb{C}^2.$ I want to understand why is $\mathbb{P}(Sym_2(\mathbb{C}^2)) \cong \mathbb{P}^2(\mathbb{C})$.

Any help please?

Answer 1

Because $\text{Sym}_2(\Bbb C^2) \cong \Bbb C^3$: The vector space of symmetric $2\times 2$ matrices is $3$-dimensional.