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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?

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    Using the [Moore-Penrose pseudoinverse](http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse), we need to figure out a linear system equivalent to $(x-PA^+b)\in P\cdot\mathrm{Ker}(A).$ I have a feeling it can be done using $PP^{T}$ and $(I-A^+A)$ but I've given up at the moment.2011-07-25

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