Q:

What's the meaning of projection with kernel $\Bbb Rv$?

In Lemma 6.13, it says "For any $v\in\Bbb R^N\setminus \Bbb R^{N-1}$, let $\pi_v:\Bbb R^N\to \Bbb R^{N-1}$ be the projection with kernel $\Bbb Rv$".
What's the meaning of projection with kernel $\Bbb Rv$?

A:

Imagine a nonvertical line $L$ through the origin in $\mathbb{R}^3$. Then one can project any point in space to a point in the $xy$-plane by following a line parallel to $L$. The kernel of that linear transformation is $L$, which can be described as the set of all multiples $\mathbb{R}v$ for any nonzero vector $v$ on $L$ - that is, any vector that does not lie in the $xy$-plane.
This is the $n=3$ case of the question. It's easy to see how it generalizes.

