Let $T$ be a linear transformation on the real vector space $R^n$ over $R$ such that $T ^2 =μT$ for same $μ∈R$ . Then which of the following is/are true?
$||Tx|| = |μ| || x||$ for all $x ∈R^n$
If $||Tx|| = || x|| $for some non zero vector $x ∈R^n$, then $μ=±1$
$T= μI$ where $I$ is the identity transformation on$R^n$
If $||Tx ||>x$ for a non zero vector $x \in R^n$, then $T$ is necessarily singular.
I am completely stuck on it. Can anybody help me please?