# Transforming a system of equations (line or any curve)?

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?

• There exists a matrix $Q$ and a vector space $K$ such that $x=Pv$ iff $v-Qx\in K$. Hence $T=\{Pv;v\in S\}$ is characterized by: $x\in T$ iff $AQx-b\in A(K)$. This translates as $m-\text{dim}A(K)$ equations. – Did Jul 25 '11 at 7:53
• Using the Moore-Penrose 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. – anon Jul 25 '11 at 9:06