Is projection of points onto plane considered a linear transformation

Question

I have a set of points in $\mathbb{R}^3$ and I need to project them onto a plane with origin $p$ and normal $n$. I can project individual point $v$ like this: $$x = v + \frac{(p-v)\cdot n}{n\cdot n}n$$ However I wonder if projection of a set of points onto line is considered a linear transformation. Such that we can use a matrix that can be calculated once? Or do we have to perform the same operation on each point individually?

Answer 1

It will not be linear if the plane does not contain the origin (since it will send $v=0$ to a nonzero point.)

If it does contain the origin, then $x = v - \frac{v \cdot n}{n \cdot n} n$ can be written as $x = Mv$ where $M = I - \frac{1}{n \cdot n} nn^\top$.