Projectors onto the same subspace but with different kernels

Question

I have two projectors $P, Q$ onto the same subspace $L \subset \mathbb{R^n}$ which are defined by their matrices $n \times n$. And it's said that there are such projectors that $\ker P \neq \ker Q$

But I can't imagine such situation and think up such projectors. Could you please help me?

Thank you!

Answer 1

You are probably thinking about orthogonal projections. Take, in the plane, two projections onto the line $y=x$: one is the orthogonal projection, whose kernel, is of course the line $y=-x$ perpendicular to $y=x$, and the other projection is a skewed projection, taking $(x,y)$ to $(x,x)$. The kernel is the $y$-axis.

It is true however that an orthogonal projection is determined by its image, because then the kernel must be the orthogonal complement.

Answer 2

Take $n=2$ and the projections given by the matrices $\left(\matrix{1 & 0\\ 0 &0}\right)$ and $\left(\matrix{1 & 1\\ 0 &0}\right)$.

The point is that not all projections are orthogonal projections.