This is a problem in algebraic geometry that I'm trying to figure out:
Problem: Let $X$ be a subvariety of $\mathbb{P}^n$ of dimension $k$, with $1 \leq k \leq n-1$. Let $P$ be a point in $\mathbb{P}^n$ and let $H$ be a hyperplane of $\mathbb{P}^n$ such that $P \notin H$. Consider the projection $\pi$ by $P$ to $H$ in $\mathbb{P}^n$. So for each $P \neq Q \in \mathbb{P}^n$, the line $\overline{PQ}$ will cut $H$ at a point $\pi(Q)$.
(i) Show that if $P \notin X$, then $\pi(X)$ is a closed subvariety of $H$. Also prove that $\dim(X) = \dim(\pi(X))$.
(ii) If $P \in X$ and there exists a point $Q \in X$ such that the line $\overline{PQ}$ is not contained in $X$, prove that the claim in (i) is still true.
Attempt: (i) I need to show that $\pi(X) \subset H$ is a closed and irreducible subset. I don't really know where to start. I let $H = Z(f)$, where $f$ is a linear polynomial. Then $f(P) \neq 0$. By applying $I$ to $\pi(X) \subset H$, I have $(f) \subset I (\pi(X))$.
I don't know how to proceed. Can I assume that $X$ is generated by $(n-k)$ amount of homogenous polynomials, because it is of dimension $k$?
How can I translate this projection to the commutative algebra side?