Is there a projection matrix for 2D to 1D perspective projection? I was wondering, if there is a projection matrix for a perspective projection of a 2D point to a line.
E.g. a random point being projected to the line at $x=1$, parallel to the y axis in the direction of the origin $(0,0)$.
I know that the easiest way to compute this would be to solve the linear equation at $x=1$ (or the intersection of the lines), but isn't there be a matrix based solution too?
To be concrete: I'm looking for a matrix $A$, that solves the equation $x \rightarrow Ax$ with $A\in \mathbb{R}^{3\times3} $


To conclude Emilios answer below, the matrix I was looking for can be written as
$$x\rightarrow Ax,\text{with }A=
\begin{bmatrix}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
\frac{1}{\text{intercept}_x} & \frac{1}{\text{intercept}_y} & 0
\end{bmatrix} 
$$
 A: If I well understand your question, the answer can be done using homogeneous coordinates. 
Given a point $P=(a,b)$, his homogeneous coordinates are $P=[a,b,1]^T\equiv [ca,cb,c]^T$ ( see here for a definition).
using this the projection from the origin on the line $x=1$ can be represented by the matrix:
$$A=
\begin{bmatrix}
1&0&0\\
0&1&0\\
1&0&0
\end{bmatrix}
$$ 
that gives:
$$
\begin{bmatrix}
1&0&0\\
0&1&0\\
1&0&0
\end{bmatrix}
\begin{bmatrix}
a\\
b\\
1
\end{bmatrix}=
\begin{bmatrix}
a\\
b\\
a
\end{bmatrix}\equiv
\begin{bmatrix}
1\\
b/a\\
1
\end{bmatrix}
$$ 

For any $P=(a,b)$, the straight line from $O$ to $P$ has equation $y=\frac{b}{a}x$, so the point $P'$ of this line with $x=1$ has coordinates $P'=(1,\frac{b}{a})$
So, in homogeneous coordinates, the two points are represented as:
$$
P=\begin{bmatrix}
a\\
b\\
1
\end{bmatrix} \qquad
P'=\begin{bmatrix}
1\\
b/a\\
1
\end{bmatrix}=\begin{bmatrix}
a\\
b\\
a
\end{bmatrix}
$$
and a simple inspection show that the matrix that transforms $P \to P'$ is the matrix $A$
