Suppose $S=\{v:Av=b\}$ is the solution set to a system of $m$ equations in $n$ variables. How do I write a system of equations that give the solution set $\{Pv:v\in S\}$, where $P$ is some $p\times n$ matrix? The new system of equations would be in $p$ variables.
Here's an example of what I'm talking about: $S$ is the solution set to $\begin{array}{l} x+y+z=2 \\ 2x+3y-z=5 \end{array}$ and $P=\left(\begin{array}{ccc} 1&-1&1 \\ 0&1&1 \end{array}\right)$. Transform each element of $S$ into new coordinates using \begin{array}{l} x' = x-y+z \\ y'=y+z \end{array} and then find the equations in x' and y' that give the transformed points.
Basically, I would like to visualize what a line looks like when it's transformed from a higher dimension into two dimensions. If I'm given a parametrized curve, then all I would have to do is substitute, but when I have an algebraic equation, then I would have to parametrize or is there some other way?