I ran into the following statement: mapping of orthogonal projection onto a plane $x+y-7z = 0$ — linear operator, but the mapping of orthogonal projection onto a plane $x+y-7z = 1$ — non-linear map. Operator from $\mathbb{R}^3$. But why? I always think, that all projections $P^2 = P$ — linear operators, no? I'm not sure, but maybe because this plane will not be a subspace?
And how can I check that some map is linear operator and some no? Only by checking the properties of linear map, like $f(x+y) = f(x) + f(y)$ and $f(\alpha x) = \alpha f(x)$?
Best Answer
The linear operator need to be at least maps the zero vector to zero vector. Since on the second plane zero vector is not on the plane then it will projected to some non zero vector. Hence the projection to the second plane is not linear.