Let $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a linear transformation defined by $T(x,y,z)=(x+y,y+z,z+x)$. I want to find two dimensional invariant subspace under $T$.
I know that two dimensional subspaces are plane passing through origin. Therefore, the two dimensional subspace $W=\{(x,y,z)\mid ax+by+cz=0\}$. If $W$ is invariant, $(x,y,z)\in W \Rightarrow T(x,y,z)\in W$. This gives $a(x+y)+b(y+z)+c(z+x)=0$. Therefore, I have the following two equations
$$ax+by+cz=0$$ and
$$ay+bz+cx=0.$$
Please suggest how to solve these two equations for $a,b,c$. Is there any better way to find two dimensional invariant subspace under $T$?