Welcome to mathimatikoi.org forum; Enjoy your visit here.

## On linear operators

Linear Algebra
Riemann
Articles: 0
Posts: 169
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

### On linear operators

Let $\alpha \in \mathbb{R} \setminus \{0\}$ and suppose that $F ,G$ are linear operators from $\mathbb{R}^n$ into $\mathbb{R}^n$ satisfying

\begin{equation*}F\circ G - G \circ F =\alpha F \end{equation*}
1. Show that for all k \in \mathbb{N} one has

$F^k \circ G - G \circ F ^k= \alpha k F^k$
2. Show that there exists k \geq 1 such that F^k =0
.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$