Let $T$ be a linear transformation on the real vector space $\mathbb R^{n}$ over $\mathbb R$ such that $T^2 = \lambda T$ for some $\lambda \in R$. Then which of the following options are true
$||Tx|| = |\lambda| ||x||$ for all $x \in R^n.$
If $\|Tx\| = \| x\| $for some non zero vector $x \in\mathbb R^n$, then $\lambda=\pm1$
$T= \lambda I$ where $I$ is the identity transformation on $\mathbb R^n$
If $\|Tx \|>\|x\|$ for a non zero vector $x \in \mathbb R^n$, then $T$ is necessarily singular.
I tried to solve by trying to find $T(T(x)) = \lambda T(x)$. Let $y= T(x)$ for all $x \in \mathbb R^n$.
Then $T(y) = \lambda y$. implies that $\lambda$ is an eigen value of $T$ and entire image of $T$ is an eigen vector corresponding to $\lambda$. I dont know whether I am right or not? and i cannot connect with the conclusions with this logic... I have searched this question in mathstack, I found one and i am not convinced with the answer given there....So before marking this question duplicate, please tell me how can i solve this question...