Answer to Question #208705 in Linear Algebra for Kok

Question

Can you construct a linear transformation T : R (3) 2 → R

3

such that

Im(T) = {(x, y,z) ∈ R

3

: ax + by + cz = 0} where a, b, c ∈ R are constants?

Answer 1

The aim is to construct a transformation $T:\mathbb{R}^3\rightarrow\mathbb{R}^3$ such that $Im(T)=\{(x,y,z)\in{\mathbb{R}}^3|ax+by+cz=0\}$. We can set: $T(x,y,z)=(u,v,w)$, where the point $(u,v,w)$ is the orthogonal projection of $(x,y,z)$ on the plane given by equation $ax+by+cz=0.$