Let $T$ be a linear transformation on the real vector space $\mathbb R^n$ over $\mathbb R$ such that $T^2 =\mu T$ for some $\mu\in\mathbb R$ . Then which of the following is/are true?
$\|Tx\| = |\mu| \|x\|$ for all $x \in\mathbb {R^n}$
If $\|Tx\| = \| x\| $for some non zero vector $x \in\mathbb R^n$, then $\mu=\pm1$
$T= \mu I$ 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 am completely stuck on it. Can anybody help me please?