5
$\begingroup$

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?

  1. $\|Tx\| = |\mu| \|x\|$ for all $x \in\mathbb {R^n}$

  2. If $\|Tx\| = \| x\| $for some non zero vector $x \in\mathbb R^n$, then $\mu=\pm1$

  3. $T= \mu I$ where $I$ is the identity transformation on $\mathbb R^n$

  4. 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?

1 Answers 1

6

Hint: Consider $T=\begin{pmatrix}\sqrt{2}&0\\0&0\end{pmatrix}$ for items 1-3. In each of item 1 and item 2, find a suitable $x$ to refute the statement. For item 4, consider $T=2I$.