This is a problem from Hoffman and Kunze (Sec 3.4, problem 10, page 96).
If T is a linear operator on $R^2$ defined by $T(x_1,x_2)=(x_1,0)$ we see that it is represented in the standard basis by the matrix A such that $A(1,1)=1$ and rest of the entires are $0$. This operator satisfies $T^2=T$. Prove that if $S$ is any linear operator satisfying $S^2=S$, then $S=0$ or $S=I$ or there is an ordered basis $\beta$ for $R^2$ such that $[S]_{\beta}= A$ (above)
Any hints to solve this problem would be greatly appreciated.
Thank you.