I 3-parameter group of symmetries on the plane

[mathwonk]

have you tried to answer this yourself? Let N be the null space of I-A, i.e. the null space of A-I. Then N is the subspace where A restricts to the identity I. In particular, A leaves this space invariant, i.e. A maps N into N. Then since A is orthogonal, if H is the ortho-complement of N, then A also leaves H invariant, i.e. for all v in H, Av is also in H. What about (I-A)v? well (I-A)v = v - Av, and both v and Av are in H, hence also their difference.

you see how easy this is? I am tempted to think you did not spend much effort trying to solve it yourself, since you are obviously smart, as you routinely catch my careless statements. Before asking such questions, do you practice spending at least 30 minutes trying to solve it as an exercise? You will soon find that you can do many of them yourself.
In fact it is clear you are already doing this to some extent, since you have suggested several corrections to my typos. keep up the good work.

 

[cianfa72]

Yes, that makes sense. Thank you